Algebra
Sequences & Series
MMTS_Full_Test_01
Grade 12
Question:
If $p,q,r$ are in AP and $p^2,q^2,r^2$ are in HP, then $p,q,r$ are
in GP
$p=q=r$
$p=q=r$ or $p,q,r$ in GP
$2p=q+r$
Step-by-Step Solution
Key Concept: AP: $2q=p+r$. HP for $p^2,q^2,r^2$: $2/q^2=1/p^2+1/r^2$
$2q=p+r$ and $\frac{2}{q^2}=\frac{1}{p^2}+\frac{1}{r^2}=\frac{p^2+r^2}{p^2r^2}$. $2p^2r^2=q^2(p^2+r^2)$. And $4q^2=(p+r)^2\Rightarrow 8p^2r^2=(p+r)^2(p^2+r^2)$... Either $p=r$ (giving $p=q=r$) or $p,q,r$ in GP.
Correct Answer: 1