Question

The fourth, seventh and tenth terms of a G.P. are p, q, r respectively, then:

• A. $p^2=q^2+r^2$
• B. $q^2=pr$
• C. $p^2=qr$
• D. $pqr+pq+1=0$

Answer 1

The correct option is B $q^2=pr$

Since, fourth and tenth terms are at equal distance from seventh term.
Hence, seventh term can be taken as the GM of fouth and tenth term.
$\therefore$ $(T_7{)}^2=T_4\times T_{1}0$
$\therefore$ $q^2=pr.$

Alternatively:
Let a be the first term and r be common ratio.
i.e., given conditions
Fourth term of G.P: $p=T_4=a{r}^3$ ...(1)
Seventh term of G.P: $q=T_7=a{r}^6$ ...(2)
Tenth term of G.P: $r=T_{10}=a{r}^9$ ...(3)
$Equ.\left(1\right)\times Equ.\left(3\right):$
$pr=a{r}^3\times a{r}^9\Rightarrow pr=a^2{r}^{12}\Rightarrow pr=(a{r}^6{)}^2\Rightarrow pr=q^2$