Question

Three numbers form a $G.P$. If the $3rd$ term is decreased by $64$, then the three numbers thus obtained will constitute an $A.P$. If the second term of this $A.P$. is decreased by $8$, a $G.P$. will be formed again. Determine the numbers.

Answer 1

Let $a,ar,a{r}^2$ are in $G.P$
Then, $a,ar,a{r}^2-64$ are in $A.P$
$\therefore 2ar=a+a{r}^2-64$,
or $a(r^2-2r+1)=64$.......(1)
Again $a,ar-8,a{r}^2-64$ are in $G.P.$
$\therefore (ar-8{)}^2=a(a{r}^2-64)$ or $-16ar+64=-64a$
or $a(16r-64)=64$........(2)
$\therefore r^2-2r+1=16r-64$ [by (1) and (2)]
or $r^2-18r+65=0$
$\therefore (r-5)(r-13)=0$ $\therefore r=5$, hence $a=4$,
$\therefore$ Number are $4,20,100$ which are in $G.P.$.