Sum to Infinity of a Geometric Series

[odolwa99]

Homework Statement

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.

Homework Equations

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

The Attempt at a Solution

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

Ans.: From textbook = 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.

 

[micromass]

Yes, 3 seems to be correct here.

 

[odolwa99]

Great. Thanks for confirming that with me.