don't know where to begin or what the question is

[anhe123]

This question I cannot answer:

For every integer K from 1 to 10, inclusive, the Kth term of a certain sequence is given by (-1)^(K+1) * (1/(2^K)). 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 1/2 and 1
d) Between 1/4 and 1/2
e) Less than 1/4

Would love a thorough explanation on it

Thanks!

[RickH]

I think I have an idea how to do this, but what are the answer choices first?

[Maciek]

Hi all!

IMO D

it is a simple example of an alternating series that converges absolutely( includes only the 10 first terms)

first term A is 1/2

common ratio R is -1/2

if this sequence were an infinite geometric series, T would be 1/3

1/2 - 1/4 + 1/8 - 1/16 + ... = (1/2)/(1 - (-1/2)) = 1/3
Source: https://en.wikipedia.org/wiki/Geometric_progression

the 2 first terms add up to 1/4 so the sum T of the ten first terms must be greater than 1/4 and less than A 1/2.

It is answer D

Hope it helps!

Best,
Maciek

[Saurav16]

Hi I think the answer is D

The sequence is an alternating sequence as you can see by the first part of the given expression (-1)^(K+1). This part of the sequence will be positive when K=1 and negative when K=2 and alternate there after. What I did was I wrote out the K's and then the sign of the first part of the expression and then evaluated the 2nd part of the expression i.e.:

K=1 : (+1)*(1/(2^1))=1/2
k=2 : (-1)*(1/(2^2))=-1/4

and so on till k=10

Eventually you get

1/2-1/4+1/8-1/16+1/32-1/64+1/128-1/256+1/512-1/1024=341/1024

so looking at it we know its a small number 341 looks like it is less than half of 1024 but lets check. So find one quarter of 1024 which equals 256. 341 > 256 but less than a 1/2 of 1024. Therefore your answer is D.

Adding them up is simple