In a certain sequence, the term $a_n$ is defined by the formula $a_n=2*a_{n-1}$ for each integer $n \ge 2$ . If $a_1=1$ , what is the positive difference between the sum of the first 10 terms of the sequence and the sum of the 11th and 12th terms of the same sequence?

Answer Choices

Correct answer marked below

1,024

1,025

2,048

2,049

Correct

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Comments

SritamApr 28

Is there a shortcut here instead of actually listing all the terms? I started writing them out and it got messy around a_8.

ArthurMay 5

Think about the sum formula for a geometric sequence — that's the cleaner route.

FelipeMay 1

Same, I listed them then realized the pattern makes it way faster.

RohitJul 8

This is a geometric sequence with r=2, so the sum of the first 10 terms is 2^10 - 1. Then the 11th and 12th terms are 2^10 and 2^11. The difference comes out clean.

YourJul 18

Ah nice, that's a much cleaner way to see it than brute force.

SrushtiJul 24

Got 2048 but I keep second-guessing whether they want the difference in that order. The wording 'sum of first 10' minus 'sum of 11th and 12th' matters.

GaganJul 27

Yeah, order matters here since it's a positive difference.

KaasApr 20

Solid 655-705 level question. The exponents are easy but setting up which terms go where is where people slip.

What this question tests

What it tests

Your ability to spot and extend arithmetic and geometric sequences and work with recursive definitions.

Common trap

Off-by-one errors in indexing (a₀ vs a₁) or confusing an arithmetic difference with a geometric ratio.