If p, q and r are in A.P, Then among the following ,correct statement is 

Let  $p=a-d,q=a,r=a+d$

A) Let $a_l,a_2,a_3\mathrm{...}inA.P$

$d_1$ is common difference

 $\therefore a_p=a_{a-d}=a_1+(a-d-1)d_1$ (p^th term)

$a_q=a_a=a_1+(a-1)d_1$ (q^th term)

$a_r=a_{a+d}=a_1+(a+d-1)d_1$ (r^th term)

$a_q-a_p=(a_1+(a-1)d_1)-(a_1+(a-d-1)d_1)=d{d}_1$

$a_r-a_q=(a_1+(a+d-1)d_1)-(a_1+(a-1)d_1)=d{d}_1$

$\therefore p^{th},q^{th}and{r}^{th}termsofA.PareinA.P$

B) Let $a_l,a_2,a_3\mathrm{...}inG.P$

 $r$ is common ratio

$\therefore a_p=a_{a-d}=a_1{r}^{a-d-1}$ (p^th term)

$a_q=a_a=a_1{r}^{a-1}$(q^th term)

$a_r=a_{a+d}=a_1{r}^{a+d-1}$(r^th term)

$\frac{a_q}{a_p}=\frac{a_1{r}^{a-1}}{a_1{r}^{a-d-1}}=r^{a-1-a+d+1}=r^d$

$\frac{a_r}{a_q}=\frac{a_1{r}^{a+d-1}}{a_1{r}^{a-1}}=r^{a+d-1-a+1}=r^d$

C) Let  $a_l,a_2,a_3\mathrm{...}inH.P$

$then\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},\mathrm{...}inA.P$

$d_1$ is common difference

$\therefore a_p=a_{a-d}=\frac{1}{\frac{1}{a_1}+(a-d-1)d_1}$(p^th term)

$a_q=a_a=\frac{1}{\frac{1}{a_1}+(a-1)d_1}$ (q^th term)

$a_r=a_{a+d}=\frac{1}{\frac{1}{a_1}+(a+d-1)d_1}$ (r^th term)

$\frac{1}{a_q}-\frac{1}{a_p}=(\frac{1}{a_1}+(a-1)d_1)-(\frac{1}{a_1}+(a-d-1)d_1)=d{d}_1$

$\frac{1}{a_r}-\frac{1}{a_q}=(\frac{1}{a_1}+(a+d-1)d_1)-(\frac{1}{a_1}+(a-1)d_1)=d{d}_1$

$\therefore p^{th},q^{th}and{r}^{th}termsofH.PareinH.P$