Question

Insert three numbers between 1 and 256 so that the resulting sequence is a GP.

Answer 1

$\text{Let the required numbers be}{G}_1,G_2,G_3.\text{then,}1,G_1,G_2,G_3,256\text{are in GP.}\text{Let r be the common ratio of this GP. Then,}{T}_5=256\Rightarrow a{r}^{(5-1)}=256\Rightarrow 1\times r^4=256\Rightarrow r^4=256=(4{)}^4\Rightarrow r=4.\therefore G_1=(1\times 4)=4,G_2=(1\times 4^2)=16and{G}_3=(1\times 4^3)=64.\text{Hence, the required numbers are 4, 16, 64.}$