Sum to infinity of the geometric progression

Question

The terms of an infinite series S are formed by adding together the corresponding terms in two infinite geometric series, T and U.

The first term of T and the first term of U are each 4.

In order, the first three terms of the combined series S are 8, 3, and 5/4.

What is the sum to infinity of S?

I have no clue on how to solve this and will be grateful for any help!

Answer 1

We have

$T_1 = 4, U_1 = 4$

$T_2 + U_2 =3$

$T_3 + U_3 = 5/4$

Also, as $T$ and $U$ are geometric series',

$T_2 = 4r_t$ $U_2 = 4r_u$

(where $r_t$ an $r_u$ are the ratios for those geometric series;)

$T_3 = 4r_t^2$ $U_3 = 4r_u^2$

So,

$4r_t + 4r_u = 3$

$4r_t^2 + 4r_u^2 = 5/4$

We now have simultaeneous equations in $r_u$ and $r_t$. You should be able to do the rest, I think.