• # question_answer In a G.P. the sum of three numbers is 14, if 1 is added to first two numbers and subtracted from third number, the series becomes A.P., then the greatest number is  [Roorkee 1973] A) 8 B) 4 C) 24 D) 16

Let three numbers in G.P. are $\frac{a}{r},\ a,\ ar$. Condition I: $\frac{a}{r}+a+ar=14\Rightarrow a\left( \frac{1}{r}+1+r \right)=14$ ?.(i) Condition II: $\frac{a}{r}+1,\ a+1$ and $ar-1$ will be in A.P., then $2(a+1)=\frac{a}{r}+1+ar-1=\frac{a}{r}(1+{{r}^{2}})$ ......(ii) From (i) and (ii), we get $a=4$ and $r=2$. So, required numbers are 2, 4, 8. Hence greatest number is 8.