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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?

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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?

Jan 24, 2015

#1
+101149
+10

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

Jan 24, 2015

#1
+101149
+10

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
+101741
+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

Jan 25, 2015