Question

For every integer k from 1 to 10, inclusive the "k"th term of a certain sequence is given by (−1)(k+1)∗(12k). If T is the sum of the first 10 terms in the sequence, then T isA. Greater than 2B. Between 1 and 2C. Between 1/2 and 1D. Between 1/4 and 1/2E. Less than 1/4

Answer 1

Option D. is the correct option.

Explanation:

In this question expression that represents the kth term of a certain sequence is not written properly.

The expression is
`(-1)^(k+1)((1)/(2^(k)))`.

We have to find the sum of first 10 terms of the infinite sequence represented by the expression given as
`(-1)^(k+1)((1)/(2^(k)))`.

where k is from 1 to 10.

By the given expression sequence will be
`(1)/(2),((-1))/(4),(1)/(8).......`

In this sequence first term "a" =
`(1)/(2)`

and common ratio in each successive term to the previous term is 'r' =
`(((-1))/(4))/((1)/(2) )`

r =
`-(1)/(2)`

Since the sequence is infinite and the formula to calculate the sum is represented by

`S=(a)/(1-r)` [Here r is less than 1]

`S=((1)/(2) )/(1+(1)/(2))`

`S=((1)/(2))/((3)/(2) )`

S =
`(1)/(3)`

Now we are sure that the sum of infinite terms is
`(1)/(3)`.

Therefore, sum of 10 terms will not exceed
`(1)/(3)`

Now sum of first two terms =
`(1)/(2)-(1)/(4)=(1)/(4)`

Now we are sure that sum of first 10 terms lie between
`(1)/(4)` and
`(1)/(3)`

Since
`(1)/(2)>(1)/(3)`

Therefore, Sum of first 10 terms will lie between
`(1)/(4)` and
`(1)/(2)`.

Option D will be the answer.