The idea is something like "What conclusion are we to draw from the doctrine previously taught?"

Keep the context of Romans in mind (see table above) as you study Romans 6, 7 and 8. In the previous three chapters Paul has focused on crucial need for and provision of justification by faith through grace (past tense salvation) but now begins the practical section on sanctification (present tense salvation) which describes saints as saved from the power of sin through the finished work of Christ. Sin in Romans 6-8 is not a reference so much to the guilt of sin (that is dealt with in Romans 1-5) but with Sin as a power in the believer's life.

S Lewis Johnson has said that in Romans 6-8...

Wrath and justification, then, yield to the discussion of sin and sanctification. Justification is restoration to life, while sanctification is restoration to health. Justification brings us from the tomb; sanctification delivers us from the old "threads." (Romans 6:1-14 PDF Format)

And when He had said these things, He cried out with a loud voice, "Lazarus, come forth." He who had died came forth, bound hand and foot with wrappings; and his face was wrapped around with a cloth. Jesus said to them, "Unbind him, and let him go." (John 11:43-44)

Ray Stedman wrote that

"Verses 1-14 of the sixth chapter of Romans are the most important fourteen verses in Scripture, insofar as being delivered from enduring the Christian life to enjoying it is concerned. There is a difference between possessing eternal life, which all Christians have, and possessing that abundant life which the Lord came to give...I saw a sign the other day and it read, "When all else fails, follow directions." That is a good sign to hang over the sixth chapter of Romans." (from his sermon entitled The Day I Died)

MacArthur writes that

"Paul moves from demonstrating the doctrine of justification, which is God’s declaring the believing sinner righteous (Ro 3:21-5:21), to demonstrating the practical ramifications of salvation on those who have been justified. He specifically discusses the doctrine of sanctification, which is God’s producing actual righteousness in the believer." (MacArthur, J.: The MacArthur Study Bible Nashville: Word Pub)

Moule explains that in this next section

"IN a certain sense, St. Paul has done now with the exposition of Justification. He has brought us on, from his denunciation of human sin, and his detection of the futility of mere privilege, to propitiation, to faith, to acceptance, to love, to joy, and hope, and finally to our mysterious but real connection in all this blessing with Him who won our peace. From this point onwards we shall find many mentions of our acceptance, and of its Cause; we shall come to some memorable mentions very soon. But we shall not hear the holy subject itself (justification) any more treated and expounded. It will underlie the following discussions everywhere; it will as it were surround them, as with a sanctuary wall. But we shall now think less directly of the foundations than of the superstructure, for which the foundation was laid. We shall be less occupied with the fortifications of our holy city than with the resources they contain, and with the life which is to be lived, on those resources, within the walls. Everything will cohere. But the transition will be marked, and will call for our deepest, and let us add, our most reverent and supplicating thought." (Moule, C. G. The Epistle of St Paul to the Romans. Ages)

Vine adds that Paul's

"aim in this chapter is to show the inconsistency of continuing in sin after being justified by grace. He makes clear that newness of life and continuance in sin are a contradiction of the new life in Christ. Chapter five constitutes the basis of the teaching of chapter six. Chapter five speaks of the means by which God has bestowed spiritual life, chapter six of how we are to live the life. The leading theme of this chapter is identification with Christ; that is the very essence of the new life. While the keynote of chapter five is “through Christ,” that of chapter six is “in Christ.”...“In Christ” (Romans 6:3, 11, 23) suggests that we are in union of life with Him in glory (He is now exalted and glorified), on the ground of what He accomplished on the Cross. Chapter six has another keynote, namely, “unto God.” That expresses how the new life is to be lived (Romans 6:10, 11, 13, 22)." (Bolding added) (Vine, W. Collected writings of W. E. Vine. Nashville: Thomas Nelson or Logos)

Note the table above which indicates that Romans 6-8 "belong together" and should not be studied independently. Therefore, although reading and digesting the truth in Romans 6 is foundation to living a life of freedom and progressive sanctification, you would be wise to study all three chapters carefully as a unit. You will note that Romans 6 deals with the believer being dead to sin; Romans 7, with the believer being dead to the Law; and Romans 8, with the believer alive in Spirit-given victory. Romans 6 tells us how sin no longer reigns over us. Romans 7 explains how the Law no longer reigns over us. And finally Romans 8 explains how the indwelling Spirit gives us life and liberty.

Wayne Barber says that...

"In Ro 6:1 the Apostle Paul has anticipated a question being asked by those who see grace as a license to sin—the Antinomians. These were the party-goers. "I’m under grace—I can do what I want to do! I’m free in Jesus—I can do what I want to do." Freedom is not the license to do what you want to do, to do what you please. It’s the power to do as you should. It’s a totally different thought. The Antinomians would take what Paul said and try to pervert it...You see, a lot of people still think, "I made a decision years ago. I walked the aisle. I cried big tears and asked God to forgive me. I’m a Christian now, and I can live like I want to live because of God’s grace. He saved me, and He forgave me." Hold it! Hold it! What were you saved from and what were you saved to? You must understand what Paul is saying here. There is no possible way a Christian can go back and live the lifestyle he lived when he was in Adam. Because he is not in Adam any more. He is now in Christ. That is the question he anticipates, and he is going to answer it." (Wayne Barber)

Wuest adds...

So Paul proposes the question, “What shall we say then?”—say then to what? We go back to Ro 5:20 for our answer which we find in the apostle’s statement, “Where sin abounded, there grace was in superabundance, and then some on top of that.”  (Paul’s teaching is that no matter how much sin committed, there are always unlimited resources of grace in the great heart of God by which to extend mercy to the sinning individual) The objector’s thought was as follows; “Paul, do you mean to tell me that God is willing to forgive a person’s sins as often as he commits them?” In response to Paul’s affirmative answer, this legalist says in effect, “Well then, if that is the case, shall we Christians keep on habitually sinning in order that God may have an opportunity to forgive us and thus display His grace?” That is the background of this man’s reasoning." (Wuest, K. S. Wuest's Word Studies from the Greek New Testament: Studies in the Vocabulary of the Greek New Testament: Grand Rapids: Eerdmans)

Bob Deffinbaugh makes a comment that

"There is a corollary to the principle that grace always outruns and exceeds sin, and it is this: sin always seeks to use that which is good to promote evil." Interesting thought! (Romans 6:1-14 An End to the Reign of Death)

Peter may have been referring to passages like this when he wrote that  in some of Paul's letters there

"are some things hard to understand, which the untaught and unstable distort (twist or dislocate the limbs on a rack = singularly graphic word applied to the perversion of scripture), as they do also the rest of the Scriptures, to their own destruction." (see note 2 Peter 2:16)

ARE WE TO (continually) CONTINUE IN SIN: epimenômen têi hamartiâi: (Ro 6:15; 2:4; 3:5-8,31; 5:20,21; Gal 5:13; 1Pe 2:16; 2Pe 2:18,19)

Bishop Moule in his classic devotional work on Romans introduces this section discussing the objection expected in many who have read and reasoned through Romans 1-5...

“We need not, then, be holy, if such is your programme of acceptance.” Such was the objection, bewildered or deliberate, which St. Paul heard in his soul at this pause in his dictation; he had doubtless often heard it with his ears. Here was a wonderful provision for the free and full acceptance of “the ungodly” by the eternal Judge. It was explained and stated so as to leave no room for human virtue as a commendatory merit. Faith itself was no commendatory virtue. It was not “a work,” but the antithesis to “works.” Its power was not in itself but in its Object. It was itself only the void which received “the obedience of the One” as the sole meriting cause of peace with God. Then — may we not live on in sin, and yet be in His favour now, and in His heaven hereafter?

Let us recollect, as we pass on, one important lesson of these recorded objections to the great first message of St. Paul. They tell us incidentally how explicit and unreserved his delivery of the message had been, and how Justification by Faith, by faith only, meant what was said, when it was said by him. Christian thinkers, of more schools than one, and at many periods, have hesitated not a little over that point. (Moule, C. G. The Epistle of St Paul to the Romans. Ages)

Paul then begins this next major section (Romans 6-8) with a rhetorical question (asked merely for effect with no answer expected). Rhetorical questions are used in fact to make statements. Paul is responding to preceding verse in which he stated that

"where sin increased, grace abounded all the more" (see note Romans 5:20).

Vincent comments that this rhetorical question is

A transition-expression and a debater’s phrase” (Morison). The use of this phrase points to Paul’s training in the Rabbinical schools, where questions were propounded and the students encouraged to debate, objections being suddenly interposed and answered. (Vincent, M. R. Word studies in the New Testament. Vol. 3, Page 1-65)

The idea of the rhetorical question is that if grace superabounds where sin abounds, ought we not to continue in it, and commit as much as possible, in order that grace might even more superabound? Why not go on sinning, so that grace may go on covering our sin? God would be getting more glory by our sinning through the covering of our sin.

Paul doesn't expect an answer per se but he does expect us to see the answer in the question. The answer is, we can't continue abiding in sin if we are genuine believers. As he will go on to explain in the next verse we can't live in sin if we died to it.

Some professing Christians think that once a person is saved, they can go on living in sin, a belief best known as  antinomianism (anti =against +  nomos = law. The antinomian then has a doctrine that is against the law of God and condones a loose view regarding the practice of sin. Scripture is quite clear that the believer is one who does not practice sin, John writing that...

No one (Greek = absolute negation) who is born (perfect tense) of God practices (present tense) sin, because His seed abides in him; and he cannot (does not have the intrinsic capability to continually) sin, because he is born (perfect tense) of God. (1John 3:9)

Another line of false reasoning an antinomian might bring up based on the doctrine of justification by faith alone apart from works is "If good works don't save, then evil works will not condemn either. Why worry about sin? Why try to live a godly life if works don't matter?"

Continue (1961) (epimeno from epí = upon, in or at + méno = abide, endure, continue, stay or remain > epí intensifies the meaning and so this word is a strengthened form of méno and gives the force of adherence to and persistence in what is referred to) means literally to tarry still, to stay at or with, to abide in, to continue in. The more common usage of epimeno in the NT is the literal picture is one abiding, of remaining on, of tarrying or staying at a place. It means to remain at or in the same place for a period of time.

Figuratively epimeno means to persist in, to persevere or to continue in an activity or state, such as sin here in Romans 6:1, in the faith in Colossians 1:23
(note), in work of teaching in 1Ti 4:16.

Epimeno is the word that John used of the determined Jewish leaders who persisted in trying to induce Jesus to contradict the law of Moses

But when they persisted (epimeno) in asking Him, He straightened up, and said to them, "He who is without sin among you, let him be the first to throw a stone at her." (John 8:7).

Thus Vine writes that epimeno "indicates persistence in what is referred to."

Epimeno is found 16 times in the (John 8:7; Acts 10:48; 12:16; 21:4, 10; 28:12, 14; Ro 6:1; 11:22-23; 1Cor 16:7-8; Gal 1:18; Phil 1:24; Col 1:23; 1 Tim 4:16) and once in the Septuagint (LXX) (Ex 12:39) and is translated: continue, 4; continued, 1; persevere, 1; persisted, 1; remain, 4; stay, 2; stayed, 3; staying, 1.

John 8:7 But when they persisted in asking Him, He straightened up, and said to them, "He who is without sin among you, let him be the first to throw a stone at her."

Acts 10:48 And he ordered them to be baptized in the name of Jesus Christ. Then they asked him to stay on for a few days.

Acts 12:16 But Peter continued knocking; and when they had opened the door, they saw him and were amazed.

Acts 21:4 And after looking up the disciples, we stayed there seven days; and they kept telling Paul through the Spirit not to set foot in Jerusalem.

Acts 21:10 And as we were staying there for some days, a certain prophet named Agabus came down from Judea.

Acts 28:12 And after we put in at Syracuse, we stayed there for three days.

Acts 28:14 There we found some brethren, and were invited to stay with them for seven days; and thus we came to Rome.

Romans 6:1 (note) What shall we say then? Are we to continue in sin that grace might increase?

Romans 11:22 (note) Behold then the kindness and severity of God; to those who fell, severity, but to you, God's kindness, if you continue in His kindness; otherwise you also will be cut off. 11:23 And they also, if they do not continue in their unbelief, will be grafted in; for God is able to graft them in again.

1 Corinthians 16:7 For I do not wish to see you now just in passing; for I hope to remain with you for some time, if the Lord permits. 8 But I shall remain in Ephesus until Pentecost;

Galatians 1:18 Then three years later I went up to Jerusalem to become acquainted with Cephas, and stayed with him fifteen days.

Philippians 1:24 (note) yet to remain on in the flesh is more necessary for your sake

Colossians 1:23 (note)  if indeed you continue in the faith firmly established and steadfast, and not moved away from the hope of the gospel that you have heard, which was proclaimed in all creation under heaven, and of which I, Paul, was made a minister.

1 Timothy 4:16 Pay close attention to yourself and to your teaching; persevere (present imperative) in these things; for as you do this you will insure salvation both for yourself and for those who hear you.

Epimeno was used to describe someone abiding in some one’s home as a guest with the idea of fellowship, cordial relations, dependence and social intercourse.

Luke uses epimeno with the literal meaning to describe the request of new converts to Peter asking "him to stay on (epimeno) for a few days." (Acts 10:48) and with the figurative meaning describing when Peter "continued (epimeno) knocking" (Acts 12:16) and of Paul and Barnabas' urging of the new converts "to continue (epimeno) in the grace of God" (Acts 13:43).

Paul uses epimeno twice in Romans 11 writing...

Behold then the kindness and severity of God; to those who fell, severity, but to you, God's kindness, if you continue (epimeno) in His kindness; otherwise you also will be cut off. And they also, if they do not continue (epimeno) in their unbelief, will be grafted in; for God is able to graft them in again. (see note Romans 11:22-23)

Epimeno  was sometimes used of a person’s purposely living in a certain place and of making it his permanent residence. How is it possible that a believer can take permanent residence in the house of sin? Wuest in fact paraphrases it this way... they "asked him to be their guest for certain days." It's as if Paul were asking can a true believer stay on as a house guest of  sin?

Furthermore, Paul's use of the present tense speaks of this abiding as continual. The idea is that the abiding under the rule and reign of sin is habitual. Paul was not speaking of a believer’s occasional falling into sin, as every Christian does at times because of the weakness and imperfection of the flesh. But he was speaking of intentional, willful sinning as an established pattern of life. He is saying that a genuine believer does  not continually live in habitual sin as when they were unsaved.

Johnson adds that Paul is not asking...

"about whether one may lapse into sin, but as Shedd says, "he cannot contentedly ‘continue in sin,’ without any resistance of it and victory over it."  (Romans 6:1-14 PDF Format)

Phillips paraphrase says

"Shall we sin to our heart's content and see how far we can exploit the grace of God?"

Or to phrase it another way...

"Shall we habitually sustain the same relationship to the sinful nature that we sustained before we were saved, a relationship which was most cordial, a relationship in which we were fully yielded to and dependent upon that sinful nature, and all this as a habit of life?”

Paul's question also has theological implications. Specifically, the question is whether there is any relationship between justification and sanctification? In other words, can a person really be saved (justified) and yet continue in the same pattern of sinfulness (lack sanctification)? Can there be a divine transaction that has no impact in the believer's life? The answer is of more than just theoretical interest, but has eternal ramifications.

Paul had already once alluded to distortion of the doctrine of the gospel of grace asking

"and why not say (as we are slanderously reported and as some affirm that we say), “Let us do evil that good may come”? Their condemnation is just." (Romans 3:8)

This argument is apparently exactly what opponents were leveling against the gospel of the grace of God, crying out

"If you could be saved just by faith alone in Christ alone, then you could go out and live in sin."

Their argument was that the gospel of grace provided not only a license to sin, but outright encouragement to do so.

Jude addressed a similar issue warning his believing readers of the need to contend earnestly for the faith because

"certain persons have crept in unnoticed (aorist tense, indicative mood = a historical reality, stealthily, literally get in by the side like gaining entrance secretly by a side door), those who were long beforehand marked out for this condemnation, ungodly (destitute of reverential awe toward God) persons who turn (present tense = continually transposing two things one of which is put in place of the other) the grace of our God into licentiousness (acknowledging no restraints, daring whatever their caprice and wanton petulance suggested) and (present tense = continually) deny (disown, disclaim connection with) our only Master (despotes = one who is the absolute owner and has uncontested power over another) and Lord, Jesus Christ." (Jude 1:4)

A famous historical instance of such perversion of the gospel of grace is found in the notorious life of the Russian monk Rasputin, who dominated the ruling family of Russia, the Romanovs, and became a very influential favorite of Czar Nicholas II. Rasputin taught the perverted "gospel" that salvation came through repeated experiencing of sin and repentance. He argued that because those who sin more require more forgiveness, those who sin with abandon will as they repent experience greater joy. Therefore, he reasoned, it was the believer’s duty to sin. In other words Rasputin's doctrine seems to have been

"The more a person sins, the more grace he will receive. So sin with gusto.”

At times this type of thinking has been intellectualized, as in the last century in James Hogg’s "Private Memoirs and Confessions of a Justified Sinner". Today this thinking is very common among those who wish to justify their sexual lifestyles.

Robertson comments that

"There are occasionally so-called pietists who actually think that God’s pardon gives them liberty to sin without penalty (cf. the sale of indulgences that stirred Martin Luther)." (Robertson, A. Word Pictures in the New Testament)

The idea expressed in Paul's question is alluded to in Voltaire famous statement that

“God will forgive; that is His ‘business.’”

W. H. Auden (widely considered among the greatest literary figures of the 20th century - in his early 20's he lived in Berlin, where he took advantage of the sexually liberal atmosphere) voiced a similar sentiment writing

"I like committing crimes. God likes forgiving them. Really the world is admirably arranged."

Oh, how Rasputin, Voltaire and Auden needed to hear and heed the truth of Romans 6!

Michael Andrus writes that

"You are not a Christian because you lead a Christian lifestyle. Rather you lead a Christian lifestyle because you are a Christian. There are tens of millions of religious people in this country of ours and countless millions in other lands, who are staking their eternity upon the notion that they are Christians because they try to lead a relatively Christian lifestyle. But if that were possible, why do you suppose the Apostle Paul, the greatest Christian theologian and missionary of all time, would have spent three chapters talking about justification before he ever got to the subject of how to live the Christian life? The simple truth is that you can imitate a Christian life, and you can fake a Christian life, but you cannot really live a Christian life until you are one." (from "A Call to Holy Living")

Ray Stedman writes...

"I heard of a man in this congregation who admitted that he was a homosexual and was living as one. He claimed that he did not need to make any change in his life because, as a Christian, his sins are forgiven." (for full sermon click  Can we Go on Sinning?)

Hendriksen describes

"Another and far more recent example from life: this man was an ardent evangelist. One of his favorite passages was taken from this very chapter of Romans, “You are not under law but under grace” (see note Romans 6:14). He spoke persuasively, drawing large crowds. However, his immediate neighbor never went to hear him. When someone asked that neighbor, “How is it that we never see you in his audience?” the answer was, “Because I happen to know that his back yard is filled with stolen property.” (Hendriksen, W., & Kistemaker, S. J. New Testament Commentary Set, 12 Volumes. Grand Rapids: Baker Book House)
 

Kenneth Wuest draws our attention to the little word "sin" writing that

"The first thing we must settle is regarding the word “sin,” does it refer here to sin as an abstraction, namely, to acts of sin committed by the believer or to the totally depraved nature still in him? A rule of Greek syntax settles the question. The definite article (Ed note: Definite article equates with the Greek word for "the") appears before the word in the Greek text. Here the article points back to a previously mentioned sin defined in its context. The reference is to sin reigning as king (see note Romans 5:21). There sin is personified since it reigns as a king. But one cannot conceive of acts of sin reigning as king in the life of a person. They are the result of some dominant factor reigning as a king. That can only be the evil nature still resident in the Christian. And here is the key to the interpretation of the entire chapter. Every time the word “sin” is used in this chapter as a noun, it refers to the evil nature in the Christian. Read the following verses and substitute the words “sinful nature” for the word “sin,” and see what a flood of light is thrown upon your understanding of this section of God’s Word (Romans 6:1, 2, 6, 10, 11, 13, 14, 16, 17, 18, 20, 22, 23)." (Bolding and color added) (Wuest, K. S. Wuest's Word Studies from the Greek New Testament: Studies in the Vocabulary of the Greek New Testament: Grand Rapids: Eerdmans)

Another way of looking at this issue is to note that in this verse “sin” is singular (in contrast to plural "sins") and does not refer to the ongoing death of specific sins which is part of our spiritual growth or sanctification. Rather the term sin in this verse refers to sin as a controlling power and as an enslaving tyrant.  Paul's point is that believers have died in relation to the power of sin.

J H Jowett has a strong warning for those who would trifle with sin writing that...

Sin is a blasting presence, and every fine power shrinks and withers in the destructive heat. Every spiritual delicacy succumbs to its malignant touch...Sin impairs the sight, and works toward blindness. Sin benumbs the hearing and tends to make men deaf. Sin perverts the taste, causing men to confound the sweet with the bitter, and the bitter with the sweet. Sin hardens the touch, and eventually renders a man "past feeling." All these are Scriptural analogies, and their common significance appears to be this--sin blocks and chokes the fine senses of the spirit; by sin we are desensitized, rendered imperceptive, and the range of our correspondence is diminished. Sin creates callosity. It hoofs the spirit, and so reduces the area of our exposure to pain. (from  The Grace Awakening)

THAT GRACE MIGHT INCREASE: hina hê charis pleonasêi (3SAAS): (Torrey's Topic Grace)

That (2443) (hína) means so that, for the purpose of and as in this sentence is usually connected with a verb in the subjunctive mood.

Grace (5485)  (charis) is God's unmerited favor (Click for more detailed discussion of amazing grace or charis)

Increase (4121) (pleonazo from pleion, comparative of polús = many, much) means to abound, to increase considerably the extent of an activity or state, with the implication of the result being an abundance...in this case that grace might be in abundance or surplus. The question is doesn't increasing in sin, set free a superfluity of grace somewhat like putting more money in circulation?

Pleonazo is aorist active subjunctive, the subjunctive with the conjunction hína expressing the purpose of continuing in sin.

William Newell writes that...

The message of simple grace, apart from all works, to the poor natural heart of man seems wholly inconsistent and impossible. "Why!" people say, "If where sin abounds grace overflows, then the more sin, the more grace." So the unbeliever rejects the grace plan. Moreover, the uninstructed Christian also is afraid; for he says, "If we are in a reign of pure grace, what will control our conscious evil tendencies? We fear such utter freedom. Put us under 'rules for holy living, 'and we can get along." Another sad fact is that some professing Christians welcome the "abounding grace" doctrine because of the liberty they feel it gives to things in their daily lives which they know, or could know, to be wrong." (Romans 6)

John Piper writes that Paul...

plays his own worst adversary in Ro 6:1. He has just said in Ro 5:20, "Where sin increased, grace abounded all the more." Now he asks, "What shall we say then? Are we to continue in sin so that grace may increase?" Here is the great objection to justification by grace through faith apart from works of the law. It seems to open the door to rampant sinning. In fact, it seems to invite more sinning because if grace is God's act to forgive and accept sinners on the basis of Christ's righteousness, not ours, then would not that grace shine all the brighter if we kept on sinning? The more sin there is, the more forgiveness there is. And the more sinning there is, the greater must be the righteousness of Christ to compensate for it. So doesn't Paul's radical teaching on justification open the door to careless living and indifference to holiness?" (For full sermon click Are We to Continue in Sin That Grace Might Increase?)

John MacArthur adds that...

Before salvation, sin cannot be anything but the established way of life, because sin at best taints everything the unredeemed person does. But the believer has no excuse to continue habitually in sin. Can he then possibly live in the same submissive relationship to sin that he had before salvation? Put in theological terms, can justification truly exist apart from sanctification? Can a person receive a new life and continue in his old way of living? Does the divine transaction of redemption have no continuing and sustaining power in those who are redeemed? Put still another way, can a person who persists in living as a child of Satan have been truly born again as a child of God? Many say yes. Paul says no, as verse 2 emphatically states." (MacArthur, J: Romans 1-8. Chicago: Moody Press)

Michael Andrus in his sermon on Romans 6 illustrates the heretical teaching of sinning more in order to receive more grace by quoting the following newspaper editorial by George Neavoll who wrote

"I am a great advocate of sin. When I explained this admittedly rather unorthodox position to several fellow parishioners at our church's coffee hour the other Sunday, they appeared...well, stricken. When I expounded on the subject in one of our editorial board meetings last week, I could tell my