Answer:
Given that,
[tex]2+(-8)+32+(-128)+.\ldots_{}[/tex]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,
[tex]t_n=2(-4)^{n-1}_{}_{}[/tex]To find the 5th term of the sequence,
Put n=5 in the above equation we get,
[tex]t_5=2(-4)^{5-1}[/tex][tex]t_5=2(-4)^4[/tex][tex]t_5=2(256)[/tex][tex]t_5=512[/tex]Since common ratio is less than 1, we get the sum of the series formula as,
[tex]S_n=\frac{a(1-r^n)}{1-r}[/tex]Substituting the values we get,
[tex]S_5=\frac{2(1-(-4)^5)}{1+4}[/tex][tex]=\frac{2(1+1024)}{5}[/tex][tex]=\frac{2(1025)}{5}[/tex][tex]=2(205)[/tex][tex]=410[/tex]The sum of the first 5 terms of the given series is 410.
Answer is: option B: 410
