How do you find the sum of the geometric series Sigma 144(-1/2)^(n-1) to n=1 to 7?

1 Answer
Oct 18, 2016

S_7=96 3/4

Explanation:

Here in given geometric series a_1=4 and r=-3

Now if the first term of geometric series is a and common ratio is r, the sum of geometric series S-n is

S_n=a+ar+ar^2+.................+ar^(n-1) and therefore

rS_n=ar+ar^2+.................+ar^(n-1)+ar^n

and subtracting first from second S_n(r-1)=a(r^n-1)

and S_n=(a(r^n-1))/(r-1) if r>1 or S_n=(a(1-r^n))/(1-r) if r<1

In the given geometric series Sigma144(-1/2)^(n-1), the first term is 144 and common ratio is -1/2

Hence S_7=(144*(1-(-1/2)^7))/(1-(-1/2))

= (144*(1-(-1/128)))/(1+1/2)

=144xx(129/128)/(3/2)

= 48cancel144xx129/128xx2/(1cancel3)

= 48xx129/128xx2

= 3cancel48xx129/(4cancel8cancel128)xxcancel2

= 387/4=96 3/4