Question

How do you find the sum of the infinite geometric series 18,12,8,...?

Answer 1

Sum of the infinite geometric series `{18,12,8,..}` is `54`

Explanation:

Sum `S_n` of a geometric series `{a,ar,ar^2,ar^3,ar^4,....}` upto `n` terms, whose first term is `a` and ratio of a term to its preceding term is `r` is given by

`a(r^n-1)/(r-1)`, when `r>1`

or `a(1-r^n)/(1-r)` when `r<1`.

When `n->oo`, `LtS_n->a/(1-r)`

Here in the series `{18,12,8,..}` `r=12/18=8/12=2/3<1`

Hence when `n->oo`, `LtS_n->18/(1-2/3)=18/(1/3)=18xx3/1=54`

Hence, sum of the infinite geometric series `{18,12,8,..}` is `54`.