A Comprehensive win!

Algebra Level 3

For an increasing Geometric Progression whose terms are integers, the following are the conditions given:

$\bullet$ Sum of first and the last term is $66$ .

$\bullet$ The product of the second and the second last term is $128$ .

Then find the number of terms in the GP.

12 7 6 9 8 5

1 solution

Md Mehedi Hasan
Dec 10, 2017

If the first term is $a$ and the ratio is $r$

then we can write $a+ar^{n-1}=66.............(i)\\ar\cdot ar^{n-2}=a^2\cdot r^{n-1}=128............(ii)$

From $(i)$ , $r^{n-1}=\dfrac{66-a}{a}..............(iii)$

From $(ii)$ , $a(66-a)=128\\a^2-66a+128=0\\a=2\quad \color{#D61F06}a\neq64$

From $(iii)$ , $r^{n-1}=32\quad \color{#D61F06}\text{since a=2}\\r^{n-1}=2^5\quad \color{#D61F06}\text{because r and n are integer}$

So $n-1=5\\\therefore n=\boxed{6}$