Question

How do you find the sum of `8(3/4)^(i-1)` from i=1 to 10?

Answer 1

`sum^10 8 (3/4)^(i-1) ~~ 30.198`
`i=1`

Explanation:

`sum^10 8 (3/4)^(i-1) = ?` .
`i=1`
The 1st term `(i=1)` is `8 (3/4)^(i-1) = 8 (3/4)^(1-1)=8*1=8`

2nd term `(i=2)` is `8 (3/4)^(i-1) = 8 (3/4)^(2-1)=8*3/4`

3rd term `(i=3)` is `8 (3/4)^(i-1) = 8 (3/4)^(3-1)=8*(3/4)^2`

4th term `(i=4)` is `8 (3/4)^(i-1) = 8 (3/4)^(4-1)=8*(3/4)^3`

The series is `8, 8*3/4, 8*(3/4)^2 , 8*(3/4)^3 ......,8*(3/4)^9`.

This is a geometric sequence with first term is `a_1=8` and

common ratio is `r=3/4=0.75` and number of terms is `n=10`

Sum formula of geometric sequence is `S_n=a_1((r^n-1)/(r-1))`

`:. S_10=8((0.75^10-1)/(0.75-1)) ~~ 30.198 :.`

`sum^10 8 (3/4)^(i-1) ~~ 30.198` [Ans]
`i=1`