Question

Can you tell me why the sum of geometric series of -3+15-75+375...,n=8 a positive number, not negative?

Answer 1

See below:

Explanation:

Let's look at the series this way:

We have 8 numbers. They start with `-3` and increase by multiplying by `-5`. I'm going to group these 8 numbers in groups of 2:

`-3, 15`
`-75, 375`
`-1875, 9375`
`-46875 , 234375`

If I add up each group of 2 numbers, I'll get a positive number:

`12`
`300`
`7500`
`187500`

And clearly if I add these 4 numbers up, I'll get a positive number.

Answer 2

The given GP has 1st term `a=-3` and common ratio `r=-5`

We know the sum of the series up to n th term is

`S_n=(a(1-r^n))/(1-r)`,where `r=< 1`

Here `S_n=(-3(1-(-5)^n))/(1-(-5))`

`=>S_n=-1/2(1-(-5)^n)`

`=>S_n=1/2((-5)^n-1)`

`(-5)^n` is positive for all even integral value of n . So when `n=8` it will be positive and hence the sum up to 8 (even) term will be positive.