How do you find if the sum of 27+18+12+... exists?

Answer 1

I assume your series is one where each term is 2/3 the term before it. All series where you multiply the previous number by a fraction converge. The formula is

n/(1-r)

where n is the first number and r is the ratio. In this case, n=27 and r = 2/3, so the sum is 27/(1/3) or 81.

Answer 2

Let S = 27+18+12+...........(i)
Then, 2/3*S = 18+12+8+.....(ii)
Subtracting (i) from (ii); we obtain:
S/3 = 27, giving S = 81.
Hence, the required sum is 81.

In general the sum of the infinite geometric series:
A + Ar+ A*r^2 + ......... for |r| < 1, is equal to A / (1-r).