Question

How do you find the sum of the geometric series `Sigma 1/4*2^(n-1)` n=1 to 5?

Answer 1

Sum of first five terms `n=1` to `n=5` of given series is `7 3/4`

Explanation:

In the geometric series `Sigma 1/4*2^(n-1)`

first term `a_1` is `a_1=1/4xx2^(1-1)=1/4xx1=1/4`

Subsequent terms are `a_2=1/4xx2^(2-1)=1/4xx2=1/2`

and `a_3=1/4xx2^(3-1)=1/4xx4=1`

Observe that the power of `2` increases by one from a term to the next term,

Hence common ration `r=2`

As sum of a geometric series `S_n`, with first term as `a_1` and common ratio as `r` is

`S_n=(a_1(r^n-1))/(r-1)`

Sum of first five terms `n=1` to `n=5` of given series is

`S_5=(1/4(2^5-1))/(2-1)=1/4xx(32-1)/1=1/4xx31=31/4=7 3/4`