For every integer \(k\) from \(1\) to \(10,\) inclusive the \(k \,th\) term of a certain sequence is given by

This topic has expert replies
Legendary Member
Posts: 1857
Joined: 14 Oct 2017
Followed by:3 members

Timer

00:00

Your Answer

A

B

C

D

E

Global Stats

For every integer \(k\) from \(1\) to \(10,\) inclusive the \(k \,th\) term of a certain sequence is given by \((-1)^{k+1}\cdot \dfrac1{2k}.\) If \(T\) is the sum of the first \(10\) terms in the sequence, then \(T\) is

A. Greater than \(2.\)

B. Between \(1\) and \(2.\)

C. Between \(\dfrac12\) and \(1.\)

D. Between \(\dfrac14\) and \(\dfrac12.\)

E. Less than \(\dfrac14.\)

Answer: D

Source: GMAT Prep

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 15565
Joined: 08 Dec 2008
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1266 members
GMAT Score:770
VJesus12 wrote:
Fri May 14, 2021 2:05 am
For every integer \(k\) from \(1\) to \(10,\) inclusive the \(k \,th\) term of a certain sequence is given by \((-1)^{k+1}\cdot \dfrac1{2k}.\) If \(T\) is the sum of the first \(10\) terms in the sequence, then \(T\) is

A. Greater than \(2.\)

B. Between \(1\) and \(2.\)

C. Between \(\dfrac12\) and \(1.\)

D. Between \(\dfrac14\) and \(\dfrac12.\)

E. Less than \(\dfrac14.\)

Answer: D

Source: GMAT Prep
List some terms to see the pattern.

We get: T = 1/2 - 1/4 + 1/8 - 1/16 + . . .
Notice that we can rewrite this as T = (1/2 - 1/4) + (1/8 - 1/16) + . . .

When you start simplifying each part in brackets, you'll see a pattern emerge. We get...
T = 1/4 + 1/16 + 1/64 + 1/256 + 1/1024

Now examine the last 4 terms: 1/16 + 1/64 + 1/256 + 1/1024
Notice that 1/64, 1/256, and 1/1024 are each less than 1/16
So, (1/16 + 1/64 + 1/256 + 1/1024) < (1/16 + 1/16 + 1/16 + 1/16)

Note: 1/16 + 1/16 + 1/16 + 1/16 = 1/4
So, we can conclude that 1/16 + 1/64 + 1/256 + 1/1024 = (a number less than 1/4)

Now start from the beginning: T = 1/4 + (1/16 + 1/64 + 1/256 + 1/1024)
= 1/4 + (a number less 1/4)
= A number less than 1/2
Of course, we can also see that T > 1/4
So, 1/4 < T < 1/2

Answer: D
Cheers,
Brent
Image

A focused approach to GMAT mastery