"A Quiz of 10 Questions Where Each Is Worth 4 Points More Than the Last" — GMAT® Worked Solution
From Episode 41 of Real GMAT® Problems (The GMAT® Strategy Podcast). For the full strategy behind sequences problems on the GMAT®, read: GMAT® Sequences: Write Every Term, Use Fractions, and Solve What's Actually Asked.
The Problem
Source: Official Guide for GMAT® Review, 11th Edition
In a certain quiz that consists of 10 questions, each question after the first is worth 4 points more than the preceding question. If the 10 questions on the quiz are worth a total of 360 points, how many points is the third question worth?
(A) 18
(B) 24
(C) 26
(D) 32
(E) 44
The Setup
Before doing any math, write what the question is asking.
We need the value of the THIRD question. Not the first. Not the smallest. The third.
Make that note big. Underline it. The most common wrong answer on this problem is the value of the first question — a clear sign that test takers solve correctly and then submit the wrong term.
The givens, written out half math half English:
- 10 questions total
- Each = previous + 4
- Sum = 360
- 3rd question = ?
The cleanest approach is to assign a variable to the first (smallest) question and write the rest of the sequence in terms of that variable.
Step 1: List the Terms in Terms of x
Let x = the value of the first question.
| Question | Value |
|---|---|
| 1st | x |
| 2nd | x + 4 |
| 3rd | x + 8 |
| 4th | x + 12 |
| 5th | x + 16 |
| 6th | x + 20 |
| 7th | x + 24 |
| 8th | x + 28 |
| 9th | x + 32 |
| 10th | x + 36 |
Writing out all 10 terms takes a few seconds and prevents counting errors later.
Step 2: Write the Sum Equation
Add the 10 terms. There are 10 copies of x, plus all the constants added together.
Sum of x's: 10x
Sum of constants: 0 + 4 + 8 + 12 + 16 + 20 + 24 + 28 + 32 + 36
Add the constants:
0 + 4 = 4
4 + 8 = 12
12 + 12 = 24
24 + 16 = 40
40 + 20 = 60
60 + 24 = 84
84 + 28 = 112
112 + 32 = 144
144 + 36 = 180
Total: 10x + 180 = 360
Step 3: Solve for x
Subtract 180 from both sides:
10x = 180
Divide both sides by 10:
x = 18
So the first question is worth 18 points.
Step 4: Find the Third Question
The third question is x + 8.
x + 8 = 18 + 8 = 26
The answer is (C).
Why This Problem Matters
About 21% of test takers miss this one. The most popular wrong answer is (A) 18 — which is the value of the first question, not the third.
That's the textbook example of "right math, wrong answer." After working through the algebra and arriving at x = 18, it is easy to spot 18 in the answer choices and feel the small surge of relief that signals "done." If "third question" wasn't visible somewhere on the scratch work, that's the answer that gets selected.
Three things to take away:
Write what's asked, big. The third question. Not the first. A clear note on the scratch work is the single most reliable defense against the (A) trap. Made the mistake once? That's why a permanent habit is worth conditioning now.
Variable-for-the-first-term is a general tool. Assigning x to the smallest term works on a wide range of sequences problems — arithmetic, geometric, problems where each term is the average of the previous two, and more. It requires almost no memorization, which makes it reliable under pressure when memorized formulas can backfire.
Write all 10 terms. It feels slow. It takes maybe 20 seconds. The payoff is that finding the third term, or summing the terms, becomes a glance rather than a recalculation. Don't fear making too many rows.
A faster shortcut exists — the sum of an arithmetic sequence is (number of terms) × (average term), and the average of an arithmetic sequence is the average of the first and last terms. That formula can solve this problem in fewer steps. It is also more brittle: small variations in problem phrasing can break it, and the variable approach almost always works. If the shortcut is solid for you, use it. Have the fundamental approach as a backup.
Previous problem: 240, 120, 60, 30… What Is the Least Term Greater Than 1?
Next problem: If K Is the Sum of the Reciprocals of the Consecutive Integers From 43 to 48 Inclusive — GMAT® Worked Solution
Back to the strategy article: GMAT® Sequences: Write Every Term, Use Fractions, and Solve What's Actually Asked