Question

The sum of three numbers in G.P is 42. If the first two numbers are increased by 2 and third is decreased by 4, the resulting form an A.P, find the middle term of the G.P

Answer 1

Let the three numbers in G.P. be $\frac{a}{r},a,ar.$
So, $\frac{a}{r}+a+ar=42.$
Now, the second condition gives $2(a+2)=(\frac{a}{r}+2)+(ar-4).$
$\Rightarrow 2a+4=\frac{a}{r}+ar-2.$
$\Rightarrow 2a+4+2=42-a$
$\Rightarrow a=12$ and $\therefore r+\frac{1}{r}=2.5$
$\Rightarrow r=2$ or $0.5$
Thus the numbers are $6,12,24$ taken in either order.