# Sum to Infinity of a Geometric Series

1. Aug 20, 2011

### odolwa99

1. The problem statement, all variables and given/known data

Q.: The numbers $\frac{1}{t}$, $\frac{1}{t - 1}$, $\frac{1}{t + 2}$ are the first, second and third terms of a geometric sequence.
Find (i) the value of t,
(ii) the sum to infinity of the series.

2. Relevant equations

S$\infty$ = $\frac{a}{1 - r}$

3. The attempt at a solution

I have already solved (i), the value of t = $\frac{1}{4}$.

Ans.: From text book = 6

Attempt at (ii): S$\infty$ = $\frac{a}{1 - r}$

a = $\frac{1}{t}$ = $\frac{1}{1/4}$ = 4

r = $\frac{U2}{U1}$ = $\frac{1}{t-1}$/ 4

$\frac{1}{1/4 - 1}$/ 4

$\frac{1}{-3/4}$/ 4

$\frac{-4/3}{4}$

$\frac{-4}{3}$($\frac{1}{4}$) = $\frac{-4}{12}$ = $\frac{-1}{3}$

Lastly,
S$\infty$ = $\frac{a}{1 - r}$ = $\frac{4}{1-(-1/3)}$

$\frac{4}{1 + 1/3}$

$\frac{4}{4/3}$ = 4($\frac{3}{4}$) = $\frac{12}{4}$ = 3

I have shown this problem on another site, and the other users seem to think that the book has the answer incorrect; with 3 being the correct value. I just wanted to post my attempt here too, to get a second opinion. Thank you.

Last edited: Aug 20, 2011
2. Aug 20, 2011

### micromass

Staff Emeritus
Yes, 3 seems to be correct here.

3. Aug 20, 2011

### odolwa99

Great. Thanks for confirming that with me.