An infinite geometric series with second term 9/4 has a sum of 81/8. Determine the possible values for the first term.

I understand the formula for an infinite geometric series is $$S=\frac{a}{1-r}$$
Given that, I need to find a and r. I thought I could find r by using $$t_n=ar^{n-1}$$ but I still do not know r or a. I would know how to do this if I was given two terms, but not when I am given only one.

Question

An infinite geometric series with second term 9/4 has a sum of 81/8. Determine the possible values for the first term.

I understand the formula for an infinite geometric series is $$S=\frac{a}{1-r}$$
Given that, I need to find a and r. I thought I could find r by using $$t_n=ar^{n-1}$$ but I still do not know r or a. I would know how to do this if I was given two terms, but not when I am given only one.

anonymous · Answer

$t_n=ar^{n-1} $  for $ n =2, \frac 94 = ar $

and $$ S=\frac{a}{1-r} \implies  81(1-r) = 8a \implies r =1- \frac 8 {81}a$$

anonymous · Answer

Substituting, $$\frac 9 4  = a - \frac 8 {81}a^2 \implies \frac 8 {81} a^2-a-\frac94 =0 $$

anonymous · Answer

http://openstudy.com/users/siddhantsharan#/updates/4fcbab87e4b0c6963ad520f6

anonymous · Answer

And all of these assuming my algebra is right :)

anonymous · Answer

Thank you!! :)

anonymous · Answer

Glad to help :)