word problem

[konan]

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+....=?

[GMATGuruNY]

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.

[ceilidh.erickson]

The GMAT does NOT test infinite sums. This is not a GMAT-like question, so you don't have to worry about it. What is the source of this question?

Please always post the source of your question, so students can assess which are GMAT-like and which are not.

[Matt@VeritasPrep]

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!

[ceilidh.erickson]

Whether the solution can be easily reached or not, my point is the same - to my knowledge, there has never been a real GMAT question that used the term "infinite series." The GMAT tends to be pretty consistent with its terminology. That makes this an un-GMAT-like question.

[Matt@VeritasPrep]

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!