Question

Consider a sequence -6, -12, 48, 24, -30, -36, 42, ... continuing in the same style. If the sum of the first n terms of the sequence is 132, what is the value of n?

• A. 11
• B. 13
• C. 18
• D. 24

Hint:

Try extending the sequence and adding it up term by term instead of assuming it behaves like a simple arithmetic progression.

Answer 1

The Correct Option is D

Step 1: Spot the pattern in the sequence.
The sequence is -6, -12, 48, 24, -30, -36, 42, and it keeps going the same way. Grouping the terms four at a time, the signs run negative, negative, positive, positive, and from the second group onward, the four numbers in a group are 6 times four consecutive integers (for instance -30, -36, 42, 48 are 6 times -5, -6, 7, 8).

Step 2: Find the sum of each group of four terms.
The first four terms, -6 - 12 + 48 + 24, add up to 54.
Every group after that, such as -30 - 36 + 42 + 48, or -54 - 60 + 66 + 72, or -78 - 84 + 90 + 96, adds up to 24 each time. This happens because the two negative numbers in a group are each 6 less in size than the two positive numbers that follow them, so most of the negative amount cancels out and a fixed net gain of 24 is left over.

Step 3: Build the running sum and test each option directly.
Adding term by term: after 4 terms the running sum is 54, after 8 terms it is 54 + 24 = 78, after 12 terms it is 102, after 16 terms it is 126, and after 20 terms it is 150.
Now check the running sum at the exact positions the options ask about, by extending the sequence out to 24 terms (-6, -12, 48, 24, -30, -36, 42, 48, -54, -60, 66, 72, -78, -84, 90, 96, -102, -108, 114, 120, -126, -132, 138, 144) and adding term by term: at n = 11 the running sum is 30, at n = 13 it is 24, at n = 18 it is -84, and at n = 24 it is 174. None of these come out to exactly 132.

Step 4: Decide the best available answer.
The running sum jumps from 126 (at n = 16) straight to 150 (at n = 20) and never touches 132 in between, and none of the listed values of n gives a sum of exactly 132 either. Comparing how far each option's sum is from 132 (30 is 102 away, 24 is 108 away, -84 is 216 away, and 174 is only 42 away), n = 24 is clearly the closest of the four. This points to an error somewhere in the original question's numbers or its answer choices, so this question is flagged for review, with n = 24 recorded as the closest available option.

Final Answer:
n = 24 is recorded here as the answer, though strictly no listed value of n produces a running sum of exactly 132.
\[ \boxed{n = 24 \text{ (flagged, no exact match found)}} \]