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
OA : B
Source : Veritas Prep
Went completely blank on this one. Experts...could you guys please help me break this question down?
5-Day Free Trial
5-day free, full-access trial TTP
Available with Beat the GMAT members only code
MORE DETAILS
Sequence
This topic has expert replies
-
GMATGuruNY
- GMAT Instructor
- Posts: 15539
- Joined: Tue May 25, 2010 12:04 pm
- Location: New York, NY
- Thanked: 13060 times
- Followed by:1906 members
- GMAT Score:790
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.
= 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.
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.
= 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128...Question:
1/2¹ + 2/2² + 3/2³ + 4/2� + 5/2� + 6/2� + 7/2�...?
= 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.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Thanks Mitch!GMATGuruNY wrote:BALLPARK.manik11 wrote: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
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 EXTREMELY small fractions)
= 2.
The correct answer is B.
I have a follow up question regarding this prompt. Why are we given this statement in the question :
"If the infinite sum 1/2^1 + 1/2^2 + 1/2^3 + 1/2^4 +...= 1". I see that you didn't use this in your solution. Am I missing something here?
Thanks!
Manik
-
GMATGuruNY
- GMAT Instructor
- Posts: 15539
- Joined: Tue May 25, 2010 12:04 pm
- Location: New York, NY
- Thanked: 13060 times
- Followed by:1906 members
- GMAT Score:790
The information in red is not needed to solve the problem.manik11 wrote: Thanks Mitch!
I have a follow up question regarding this prompt. Why are we given this statement in the question :
"If the infinite sum 1/2^1 + 1/2^2 + 1/2^3 + 1/2^4 +...= 1". I see that you didn't use this in your solution. Am I missing something here?
Thanks!
Manik
That said, I've amended my post to show how this information might prove helpful to some test-takers.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
-
DavidG@VeritasPrep
- Legendary Member
- Posts: 2663
- Joined: Wed Jan 14, 2015 8:25 am
- Location: Boston, MA
- Thanked: 1153 times
- Followed by:128 members
- GMAT Score:770
One alternative:manik11 wrote:
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
We're asked to find the value of: 1/2¹ + 2/2² + 3/2³ + 4/2� ... =1/2 + 2/4 + 3/8 + 4/16..
Call the series we're trying to solve 'x'. So 1/2 + 2/4 + 3/8 + 4/16... = x
If we multiply that series by (1/2) we get 1/4 + 2/8 + 3/16... = (1/2)x
Subtract the second series from the first to get [1/2 + 2/4 + 3/8 + 4/16...] - [1/4 + 2/8 + 3/16...] = 1/2 + 1/4 + 1/8 +1/16 ...
Put another way: x - (1/2)x = 1/2 + 1/4 + 1/8 + 1/16 ...
Note that we're told that 1/2 + 1/4 + 1/8 + 1/16... = 1
So x - (1/2)x =1
(1/2)x = 1
x = 2
-
Matt@VeritasPrep
- GMAT Instructor
- Posts: 2630
- Joined: Wed Sep 12, 2012 3:32 pm
- Location: East Bay all the way
- Thanked: 625 times
- Followed by:119 members
- GMAT Score:780
Here's my approach.
We're told that
1/2¹ + 1/2² + 1/2³ + 1/2� + ... = 1
We're asked for
1/2¹ + 2/2² + 3/2³ + 4/2� + ...
which is really
(1/2¹ + 1/2² + 1/2³ + 1/2� + ...) + (1/2² + 1/2³ + 1/2� + ...) + (1/2³ + 1/2� + ...) + (1/2� + ...)
We know that the part in red = 1.
We know that the part in green = the part in red - 1/2
= 1 - 1/2 = 1/2.
We know that the part in blue = the part in green - 1/4
= 1/2 - 1/4 = 1/4.
From here, you can see that we suddenly have 1 + 1/2 + 1/4 + ...,
which is just 1 + (1/2 + 1/4 + ...),
or 1 + (the part in red),
or 1 + 1, or 2.
We're told that
1/2¹ + 1/2² + 1/2³ + 1/2� + ... = 1
We're asked for
1/2¹ + 2/2² + 3/2³ + 4/2� + ...
which is really
(1/2¹ + 1/2² + 1/2³ + 1/2� + ...) + (1/2² + 1/2³ + 1/2� + ...) + (1/2³ + 1/2� + ...) + (1/2� + ...)
We know that the part in red = 1.
We know that the part in green = the part in red - 1/2
= 1 - 1/2 = 1/2.
We know that the part in blue = the part in green - 1/4
= 1/2 - 1/4 = 1/4.
From here, you can see that we suddenly have 1 + 1/2 + 1/4 + ...,
which is just 1 + (1/2 + 1/4 + ...),
or 1 + (the part in red),
or 1 + 1, or 2.
