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Hint: Use the nth term formula and proceed as instructed in question.

Complete step-by-step answer:

It has been given

$

a + a{r^{n - 1}} = 66 \\

{\text{And }} \\

ar.{a^{n - 2}} = {a^2}{r^{n - 1}} = 128 \\

\therefore {a^{n - 1}} = \dfrac{{128}}{a} \\

{\text{Putting in 1, we get }}a + \dfrac{{128}}{a} = 66 \\

\therefore {a^2} - 66a + 128 = 0 \\

{\text{splitting the middle terms we get}} \\

(a - 2)(a - 64) = 0 \\

\therefore a = 2,64 \\

{r^{n - 1}} = 32,\dfrac{1}{{32}} \\

{\text{We reject the second value as r > 1 ,}}\therefore {{\text{r}}^{n - 1}} = 32 \\

{\text{Sum = }}\dfrac{{a({r^n} - 1)}}{{r - 1}} = 126{\text{ }} \Rightarrow \dfrac{{2(32r - 1)}}{{r - 1}} = 126 \\

\because {r^{n - 1}} = 32 \\

\therefore 32r - 1 = 63r - 63 \\

\therefore r = 2{\text{ and }}{r^{n - 1}} = 32{\text{ gives}} \\

{{\text{2}}^{n - 1}} = {2^5} \Rightarrow n - 1 = 5 \Rightarrow n = 6 \\

$

Note: Must remember all the general terms, sum formula, quadratic roots in order to solve such similar problems. Similar questions can be asked for Harmonic Progression and AGP etc.

Complete step-by-step answer:

It has been given

$

a + a{r^{n - 1}} = 66 \\

{\text{And }} \\

ar.{a^{n - 2}} = {a^2}{r^{n - 1}} = 128 \\

\therefore {a^{n - 1}} = \dfrac{{128}}{a} \\

{\text{Putting in 1, we get }}a + \dfrac{{128}}{a} = 66 \\

\therefore {a^2} - 66a + 128 = 0 \\

{\text{splitting the middle terms we get}} \\

(a - 2)(a - 64) = 0 \\

\therefore a = 2,64 \\

{r^{n - 1}} = 32,\dfrac{1}{{32}} \\

{\text{We reject the second value as r > 1 ,}}\therefore {{\text{r}}^{n - 1}} = 32 \\

{\text{Sum = }}\dfrac{{a({r^n} - 1)}}{{r - 1}} = 126{\text{ }} \Rightarrow \dfrac{{2(32r - 1)}}{{r - 1}} = 126 \\

\because {r^{n - 1}} = 32 \\

\therefore 32r - 1 = 63r - 63 \\

\therefore r = 2{\text{ and }}{r^{n - 1}} = 32{\text{ gives}} \\

{{\text{2}}^{n - 1}} = {2^5} \Rightarrow n - 1 = 5 \Rightarrow n = 6 \\

$

Note: Must remember all the general terms, sum formula, quadratic roots in order to solve such similar problems. Similar questions can be asked for Harmonic Progression and AGP etc.

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