The first two terms of a geometric progression are 27 and 9. How many terms are added up to get a sum of 364/9?

Guest Jan 24, 2015

#1**+10 **

Best Answer

We are trying to solve

27*∑ (1/3)^(n-1) from 1 to n = 364/9

WolframAlpha shows that 6 terms are needed

Here are the sums from n = 1 to n = 6

27 + 9 + 3 + 1 + 1/3 + 1/9 = 364/9

CPhill
Jan 24, 2015

#2**+5 **

a=27 r=1/3

27+9+3+1+(1/3)+(1/9)=364/9

$$\boxed{S_n=\frac{a(1-r^n)}{1-r}}\\\\\\

\frac{364}{9}=\frac{27(1-(1/3)^n)}{1-(1/3)}}\\\\

\frac{364}{9*27}=\frac{1-(1/3)^n}{2/3}}\\\\

\frac{364*2}{9*27*3}=1-(1/3)^n\\\\

\frac{728}{729}-1=-(1/3)^n\\\\

\frac{-1}{729}=-\frac{1}{3^n}\\\\

729=3^n\\\\

log_3(729)=log_3 3^n\\\\

n=log_3(729)\\\\

n=log729/log3\\\\

n=6$$

Thank you Chris for helping me spot my previous error

Melody
Jan 25, 2015