How do you find the sum of the infinite geometric series 16 + 24 + ... + 81 + 121.5?
1 Answer
The infinite geometric series
The finite geometric series
Explanation:
The common ratio is
An infinite geometric series only converges if
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)