Question

If p, q, r are in A.P., then $p^{th},q^{th}$ and $r^{th}$ terms of any G.P. are themselves in

• A. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

Answer: (B)

p, q and r are in A.P

i.e. $2q=p+r$

General term of G.P is $T\left(K\right)=a{r}^{(k-1)}$

$p^{th}$ term is :- $T\left(p\right)=a{r}^{(p-1)}...\left(1\right)$

$q^{th}$ term is :- $T\left(q\right)=a{r}^{(q-1)}...\left(2\right)$

$r^{th}$ term is :- $T\left(r\right)=a{r}^{(R-1)}...\left(3\right)$

To prove that ;- $T\left(p\right),T\left(q\right),T\left(r\right)$ are in G.P

$T(q{)}^2=T(p).T(r)$

${\left[a{r}^{(q-1)}\right]}^2=a{r}^{(p-1)}.a{r}^{(R-1)}$

$a^{2}a{r}^{2(q-1)}=a^2{r}^{(p-1)}{r}^{(R-1)}$

$r^{2(q-1)}=r^{(P+R-2)}$

Taking only powers

$2(q-1)=P+R-2$

$2q-2=P+R-2$

$2q=P+R$

This proves that $p^{th},q^{th}$ and $r^{th}$

term are in G.P