108 Explanation: In any geometric series, we need two pieces of information to find its sum to the n -th term, and thus the sum to infinity. Firstly, we need the first term, a . This is obviously 36 in this expression. Secondly, we need the common ratio, r ,. We find this by taking 24/36 =2/3 Thus a =36 and r =2/3 . also note that (R) =2/3 <1
We then apply the formula for sum of geometric series to the n -th term when (r) <1 : Sn =a(1 -rn ) / 1-r = 36 ( 1 - (2/3 ) n) / 1 -(2/3) =108 (1 -(2/3) n) The sum to infinity s ♾ is defined to be the limit of sn as n - ♾ N- ♾,(2/3) n -0.sn -108 -0 =108 S ♾ =108
