Respuesta :
Answer:
Option D. is the correct option.
Step-by-step explanation:
In this question expression that represents the kth term of a certain sequence is not written properly.
The expression is [tex](-1)^{k+1}(\frac{1}{2^{k}})[/tex].
We have to find the sum of first 10 terms of the infinite sequence represented by the expression given as [tex](-1)^{k+1}(\frac{1}{2^{k}})[/tex].
where k is from 1 to 10.
By the given expression sequence will be [tex]\frac{1}{2},\frac{(-1)}{4},\frac{1}{8}.......[/tex]
In this sequence first term "a" = [tex]\frac{1}{2}[/tex]
and common ratio in each successive term to the previous term is 'r' = [tex]\frac{\frac{(-1)}{4}}{\frac{1}{2} }[/tex]
r = [tex]-\frac{1}{2}[/tex]
Since the sequence is infinite and the formula to calculate the sum is represented by
[tex]S=\frac{a}{1-r}[/tex] [Here r is less than 1]
[tex]S=\frac{\frac{1}{2} }{1+\frac{1}{2}}[/tex]
[tex]S=\frac{\frac{1}{2}}{\frac{3}{2} }[/tex]
S = [tex]\frac{1}{3}[/tex]
Now we are sure that the sum of infinite terms is [tex]\frac{1}{3}[/tex].
Therefore, sum of 10 terms will not exceed [tex]\frac{1}{3}[/tex]
Now sum of first two terms = [tex]\frac{1}{2}-\frac{1}{4}=\frac{1}{4}[/tex]
Now we are sure that sum of first 10 terms lie between [tex]\frac{1}{4}[/tex] and [tex]\frac{1}{3}[/tex]
Since [tex]\frac{1}{2}>\frac{1}{3}[/tex]
Therefore, Sum of first 10 terms will lie between [tex]\frac{1}{4}[/tex] and [tex]\frac{1}{2}[/tex].
Option D will be the answer.
