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

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

by VJesus12 » Fri May 14, 2021 2:05 am

00:00

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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 $$\dfrac14.$$

Source: GMAT Prep

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### Re: For every integer $$k$$ from $$1$$ to $$10,$$ inclusive the $$k \,th$$ term of a certain sequence is given by

by [email protected] » Fri May 14, 2021 5:30 am
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 $$\dfrac14.$$

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