Question

In an increasing geometric progression, the sum of the first term and the last term is $66$, the product of the second terms from the beginning and the end is $128$ and sum of all terms is $126$. Then the number of terms in the progression is

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is B $6$

Let $a,ar,a{r}^2,\dots ,a{r}^{n-1}$ be the increasing G.P.
From given conditions,
$a+a{r}^{n-1}=66\cdots \left(1\right)$
$ar\cdot a{r}^{n-2}=a^2{r}^{n-1}=128\cdots \left(2\right)$
$\frac{a(r^n-1)}{r-1}=126\cdots \left(3\right)$

From $\left(1\right)$ and $\left(2\right)$,
$a+\frac{128}{a}=66$
$\Rightarrow a^2-66a+128=0$
$\Rightarrow (a-64)(a-2)=0$
$\Rightarrow a=64$ or $a=2$

Case $1:$
If $a=2$, then
$r^{n-1}=32$
and $r^n-1=63(r-1)$
Solving above equations, we get
$r=2$
$\Rightarrow 2^{n-1}=32$
$\Rightarrow n=6$

Case $2:$
If $a=64$, then
$r^{n-1}=\frac{1}{32}$
$r^n-1=\frac{63}{32}(r-1)$
Solving above equations, we get
$r=\frac{1}{2}$ and $n=6$
But here, since $a>0$ and $0<r<1$
so, it is a decreasing G.P.