Two integers A and B are such that $A = 2^{65}$ and $B = 2^{64} + 2^{63} + 2^{62} + \cdots + 2^0$ . Find the relation between A and B.

Answer Choices

Correct answer marked below

A and B are equal

B is greater than A by 1

A is greater than B by 1

Correct

B is 2^{64} greater than A

A is 2^{64} greater than B

See the full step-by-step explanation

You can see the correct answer above. Sign in for free to unlock the complete worked solution.

Track your performance and improve

Get detailed analytics, unlock full explanations, and move up difficulty tiers as you practice.

Start for free Sign in

Explanation

Unlock the full explanation

Create a free account to reveal the correct answer, see the step-by-step explanation, and start tracking your GMAT progress.

Create free account Sign in

Comments

RosanneAug 16

Wait, B is a geometric series right? Sum from 2^0 to 2^64... that's 2^65 - 1. So B = 2^65 - 1 and A = 2^65, meaning A is bigger by 1. Took me a sec to see the pattern, but once you recognize the sum it's straightforward.

RadhikaAug 19

Yeah exactly, the formula is 2^(n+1) - 1 for sum up to 2^n. Classic trap though, I initially thought B might be larger.

TanveerAug 27

Same here, I almost picked B before realizing the sum formula gives one less than the next power.

ValentinaApr 16

This is a 705+ question? Seems more like a 600-level pattern recognition thing. Maybe I'm missing a trap...

ShubhangiApr 21

I think the difficulty comes from having to recall the geometric series sum formula quickly under time pressure. But yeah, if you know it, it's not that bad.

BhavyaApr 23

Can someone explain why the sum 2^64 + ... + 2^0 equals 2^65 - 1? I get that it's a geometric series but I always mess up the formula.

DibyashreeApr 27

Multiply both sides by 2 and subtract, or just use a = 1, r = 2, n = 65 terms. The result is (2^65 - 1)/(2-1).

RosaMay 2

Or think of it like binary: 111...1 (65 ones) is 2^65 - 1.

SarthakJun 12

The choices are tricky because D and E bring in 2^64, which is way bigger than 1. If you don't simplify B, you might think the difference is huge. Good question to test if you actually simplify.

What this question tests

What it tests

Your grasp of exponent rules — multiplying and dividing powers, power-of-a-power, and negative/zero exponents.

Common trap

Adding exponents when multiplying bases that have the same exponent, or confusing (x^a)^b with x^(a·b).

Two integers A and B are such that A=265A = 2^{65}A=265 and B=264+263+262+⋯+20B = 2^{64} + 2^{63} + 2^{62} + \cdots + 2^0B=264+263+262+⋯+20. Find the relation between A and B.