How do you find the sum of the infinite geometric series 16 + 24 + ... + 81 + 121.5?

1 Answer
Feb 23, 2016

The infinite geometric series #16+24+...# does not converge and therefore its sum is infinite.
The finite geometric series #16+24+...+81+121.5# has a sum of #332.5#

Explanation:

The common ratio is #r=24/16= 1.5#

An infinite geometric series only converges if #abs(r) < 1#.

However if the question meant to ask for the finite series:

If #a_0=16=2^4# and #r=1.5# #a_n=121.5# for some value of #n#
then #n=5#
This is true since for a geometric series #a_i=a_0*r^i#
and #2^4*1.5# will be a whole number if #i < 5# and will contain a fraction less than #0.5# if #i > 5#

The sum of a finite geometric series is given by the expression:
#color(white)("XXX")Sigma_(i=0)^n a_i = a_0*((1-r^(n+1))/(1-r))#

For the given series this becomes:
#color(white)("XXX")16*((1-1.5^6)/(1-1.5))= 332.5# (yes; I used a calculator)