Question

If AM between $p^{th}$ and $q^{th}$ terms of an AP be equal to the AM between $r^{th}$ and $s^{th}$ term of the AP, then $p+q$ is equal to

• A. $r+s$
• B. $\frac{r-s}{r+s}$
• C. $\frac{r+s}{r-s}$
• D. $r+s+1$

Answer 1

The correct option is A $r+s$

We know A.P formula for nth terms with 'a' as the first term and 'd' as the common difference as shown below:

$t_n=a+(n-1)d$

Also AM is given between two numbers a and b.

$A=\frac{a+b}{2}$

So arithmetic mean of pth and qth terms of AP is as shown below:

$=\frac{a+(p-1)d+a+(q-1)d}{2}$

Similarly we can have AM of rth term and sth term of AP as shown below:

$=\frac{a+(r-1)d+a+(s-1)d}{2}$

Applying the given conditions we get,

$\frac{a+(p-1)d+a+(q-1)d}{2}=\frac{a+(r-1)d+a+(s-1)d}{2}$

$\frac{a+pd-d+a+qd-d}{2}=\frac{a+rd-d+a+sd-d}{2}$

$a+pd-d+a+qd-d=a+rd-d+a+sd-d$

$2a+d(p+q)-2d=2a+d(r+s)-2d$

$d(p+q)=d(r+s)d$

$p+q=r+s$

Hence option A is correct.