Question

30 points! I don't know where to start here if I'm being honest. Please explain your answer in detail?

Answer 1

The sum of the first six terms is

`32-8+2-0.5+(1)/(8)-(1)/(32)=25.59`

Explanation:

Given series is 32-8+2-0.5+...

We may write
`{\{32,-8,2,-0.5,...}\}`

Let
`a_(1)=32,a_(2)=-8,a_(3)=2,a_(4)=-05,...`

Common ratio
`r=(a_(2))/(a_(1))`

`r=(-8)/(32)`

`r=(-1)/(4)`

`r=(a_(3))/(a_(2))`

`r=(2)/(-8)`

`r=(-1)/(4)`

Therefore the common ratio is
`r=(-1)/(4)`

Therefore given sequence is of the form of Geometric sequence

The nth term of the geometric sequence is

`a_(n)=ar^(n-1)`

First to find the 5th and 6th term

That is substitute n=5 and n=6 ,a=32 and
`r=(-1)/(4)` in above equation we get

`a_(5)=32((-1)/(4))^(5-1)`

`a_(5)=32((-1)/(4))^(4)`

`a_(5)=32[((-1)/(4))* ((-1)/(4))* ((-1)/(4))* ((-1)/(4))]`

`=(1)/(8)`

Therefore
`a_(5)=(1)/(8)`

`a_(6)=32((-1)/(4))^(6-1)`

`a_(6)=32((-1)/(4))^(5)`

`a_(6)=32[((-1)/(4))* ((-1)/(4))* ((-1)/(4))* ((-1)/(4))* ((-1)/(4))]`

`=(-1)/(32)`

Therefore
`a_(6)=-(1)/(32)`

Therefore the sum of the first six terms is

`32-8+2-0.5+(1)/(8)-(1)/(32)`

`=25.5+(3)/(32)`

`=(819)/(32)`

`=25.59`

Therefore the sum of the first six terms is

`32-8+2-0.5+(1)/(8)-(1)/(32)=25.59`