"In an Increasing Sequence of 10 Consecutive Integers..." — GMAT® Worked Solution

Source: Official Guide for GMAT® Review, 11th Edition

In an increasing sequence of 10 consecutive integers, the sum of the first 5 integers is 560. What is the sum of the last 5 integers in the sequence?

(A) 585

(B) 580

(C) 575

(D) 570

(E) 565

Try it before reading on.

Step 1: Establish What You Know and What You Don't

Before writing equations, separate the knowns from the unknowns:

• Known: 10 consecutive integers, sum of first 5 = 560
• Unknown: sum of last 5
• Unknown: the individual numbers themselves

When you identify what you don't know, you can create variables for those unknowns and use algebra to solve for them.

Step 2: Express All Terms With One Variable

Consecutive integers differ by 1. If the first integer is $a$, then the next four are $a+1$, $a+2$, $a+3$, and $a+4$.

The first five integers:

The last five integers:

One variable. Ten terms. The algebra is about to get a lot cleaner than if you'd used ten separate variables.

Step 3: Build the Equation for the First Five

The sum of the first five equals 560:

Combine like terms:

Step 4: Solve for $a$

Subtract 10 from both sides:

Divide both sides by 5:

    110
   -----
5 ) 550
    5
    ---
     50
     50
    ---
      0

The first integer in the sequence is 110.

Step 5: Find the Sum of the Last Five

The last five integers are $a+5$, $a+6$, $a+7$, $a+8$, and $a+9$. Their sum:

Plug in $a = 110$:

The answer is (A).

Alternatively, you could list the actual numbers — 115, 116, 117, 118, 119 — and add them directly. Both methods work. The algebraic approach is more generalizable: it works the same way whether the sequence has 10 terms or 100.

Why This Problem Matters

About 20% of test takers miss this one. The wrong answers are clustered tightly — all within 5 of each other — which suggests two categories of error:

Setup errors: If the algebraic expression for the consecutive integers is off by even one, the final sum shifts by 5. Picking (B) 580 instead of 585 probably means the last five were expressed as $a+4$ through $a+8$ instead of $a+5$ through $a+9$. One index off, and the answer is one choice away.

Brute-force arithmetic errors: Some test takers solve for $a = 110$, then list out all ten numbers and add the last five by hand. That works, but adding five three-digit numbers under time pressure introduces carrying errors. The algebraic sum $5a + 35$ is faster and less error-prone.

The single-variable method works because it scales. Change the problem to 100 consecutive integers and the approach is the same — $a$ through $a+99$, with sums expressed in terms of $a$. You don't need a different strategy for each variant of consecutive integer question. One system, applied consistently.

Want the full strategy behind this problem? Read: GMAT® Quant: Three Systems for Percent, Integer, and Sequence Problems

From Episode 24 of Real GMAT® Problems (The GMAT® Strategy Podcast).

Ready for the next one? "A Necklace Is Made by Stringing N Individual Beads..." — GMAT® Worked Solution.