Question

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

Answer 1

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)