Question

I need help with this asap please

Answer 1

Given that,

`2+(-8)+32+(-128)+.\ldots_{}`

To find the sum of the first 5 terms.

First, to find the first 5 terms of the given sequence.

The given sequence is 2,-8,32,-128,...

It follows geometric series with initial term 2, and common ratio as -4

The explicit formula of the given sequence is,

`t_n=2(-4)^(n-1)_{}_{}`

To find the 5th term of the sequence,

Put n=5 in the above equation we get,

`t_5=2(-4)^(5-1)`
`t_5=2(-4)^4`
`t_5=2(256)`
`t_5=512`

Since common ratio is less than 1, we get the sum of the series formula as,

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

Substituting the values we get,

`S_5=(2(1-(-4)^5))/(1+4)`
`=(2(1+1024))/(5)`
`=(2(1025))/(5)`
`=2(205)`
`=410`

The sum of the first 5 terms of the given series is 410.

Answer is: option B: 410