Find the sum of the infinite geometric series if it has a sum

Question

∑8   12 (1/3)^ n-1

n=1

Answer 1

Any geometric series where the common ratio between terms is between -1 and 1 does converge to a sum.

 

In this case, the ratio is 1/3, because each term is one higher power of 1/3 than the one before it.  That means that each term is 1/3 of the one before, which is exactly what is meant by the "common ratio."  Since 1/3 is between -1 and 1, the series does have a sum.

 

The formula for the sum is a / (1 - r), where a is the first term and r is the ratio.  In this case, the work is to find the first term.

 

According to the summation symbol, the series starts at n=1.  So plug n=1 into the expression being summed:

 

12 (1/3)^n-1 = 12 (1/3)^1-1 = 12(1/3)⁰ = 12(1) = 12.

 

So the sum of the series is 12 / (1 - 1/3) = 12 / (2/3) = 18.