Question

A group of friends have some money which was in an increasing GP. The total money with the first and the last friend was Rs $66$ and the product of the money that the second friend had and that the last but one friend had was Rs $128$. If the total money with all of them together was Rs $126$, then How many friends were there?

• A. $6$
• B. $5$
• C. $3$
• D. Can not be determined

Answer 1

The correct option is A $6$

Working backwards from the options given, we gave the solution as follows.

Let us assume the number of friends to be 6.

Then they can be written as $a,ar,a{r}^2,a{r}^3,a{r}^4,a{r}^5$

Given

$a+a{r}^5=a(1+r^5)=66$

$ar\left(a{r}^4\right)=a^2{r}^5=128$

$a\left(\frac{r^6-1}{r-1}\right)=126$

$a=\frac{66}{1+r^5}$

$a=\frac{66}{1+\left(128/a^2\right)}⟹66a=a^2+128⟹a^2+128-66a=0⟹a=2,64⟹r=2$

If a equals r equals 2, then the GP will be 2,4,8,16,32,64.

SO, there were 6 friends. Option A