Question

Find the sum of the finite geometric series a+ar+ar^2+.....+ar^n-1
1)1+2+4+8+.....+2^9
2)a=30,r=1,n=100

Answer 1

1) 1+2+4+8+................+2^9
a = 1, r => 2/1 = 2
$X_{n}$ = 2^9
2^9 = a*r^(n-1)
put the value of a and r
2^9 = 1*2^(n-1)
2^9 = 2^(n-1)
as the base is same hence power also same
9 = n-1
n = 10
$S_{n}$ = a{(r^n - 1) / (r - 1)}
put the value of a, r and n
$S_{n}$ = 1*{(2^10 - 1) / (2 - 1)}
                         = 2^10 - 1
$S_{n}$ = 1023

2) sum of G.P. is only define for r<1 or r>1 not for r=1 because if we put r = 1 then we get 0 in the denominator 
so for the sum we just add 30, 100 times
so the sum = 30*100 = 3000