finding the sum of the terms of an infinite geometric sequen

finding the sum of the terms of an infinite geometric sequence.

find each sum that converges 18 + 6 + 2 +2/3 + ...

Answer is 27.

Solution

18 + 6 + 2 + 2/3 + ...

a = 18

r = 6/18 = 1/3

S_n = a ( 1- rⁿ ) / (1-r)

S_n = 18 (1 - (1/3)ⁿ / (1- 1/3) = 3*18* (1-(1/3)ⁿ)/2 = 27 (1 -(1/3)ⁿ)

The sum converges at 27