word problem

When you post a problem, please include the answer choices.

If the infinite sum 1/2^1 + 1/2^2 + 1/2^3 + 1/2^4 +...= 1, what is the value of the infinite sum 1/2^1 + 2/2^2 + 3/2^3 + 4/2^4+....?
A) 1
B) 2
C) 3
D) π (pi)
E) Infinite

Given:
1/2¹ + 1/2² + 1/2³ + 1/2� + ... = 1

1/2 + 1/4 + 1/8 + 1/16 + ... = 1

15/16 + (sum of increasingly small fractions) = 1

(almost 1) + (sum of increasingly small fractions) = 1.

Implication:
The value in orange is just enough to bring the sum of the lefthand side to 1.

Question:
1/2¹ + 2/2² + 3/2³ + 4/2� + 5/2� + 6/2� + 7/2�...?

= 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128...

= 1/2 + 1/2 + 3/8 + 2/8 + 5/32 + 3/32 + 7/128...

= 1 + 5/8 + 8/32 + 7/128...

= 1 + 5/8 + 1/4 + 7/128...

= 1 + (almost 1) + (sum of increasingly small fractions)

= 1 + 1

= 2.

The correct answer is B.

Nah, this question is fine: it doesn't require any calculus (or even algebra II) and it rewards algebraic ingenuity. As Mitch showed, it can also be relatively easily approximated, though at the cost of a little extra time, so everything about it seems very GMATy to me, if not at first blush.

Anyway, we're told that

(1/2) + (1/2)² + (1/2)³ + (1/2)� + ... = 1

Notice that

(1/2) + (2/2²) + (3/2³) + (4/2�) + ...

is really

(1/2) + (1/2)² + (1/2)³ + (1/2)� + ...
+ (1/2)² + (1/2)³ + (1/2)� + ...
+ (1/2)³ + (1/2)� + ...
+ (1/2)� + ...

or

1 +
(1 - 1/2) +
(1 - 1/2 - 1/4) +
(1 - 1/2 - 1/4 - 1/8) + ...

or

1 +
1/2 +
1/4 +
1/8 + ...

or

1 + (1/2 + 1/4 + 1/8 + ...)

or

1 + 1

Piece of cake!

Sure, but you don't need to know that term to solve the problem, nor do you need any algebra II, let alone calculus.

As for problems being un-GMAT-like ... I can think of a couple on my exam that look like nothing I've seen in the GMAT literature, and which I would assume were out of scope if I hadn't seen them on the actual exam. You never know!