Question

If the ratio of sums of p and q terms of an AP is :p^2:q^2;then find ratio of its pth and qth terms.

Answer 1

hope it helps you a lot

Answer 2

Answer:

Ratio of pth term and qth term is 2p - 1 : 2q - 1.

Step-by-step explanation:

Given: Ratio of Sum of pth term and qth term of an AP = p² : q²

To find: ratio of the pth term and qth term.

let, a be the first term of the AP and d be the common difference of the AP

We know that, Sum of nth term of AP is given as follows:

$S_n=\frac{n}{2}(2a+(n-1)d)$

Also, nth term of AP is given as follows:

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

We have,

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

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

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

$p(2a+(q-1)d)=q(2a+(p-1)d)$

$2ap+p(q-1)d=2aq+q(p-1)d$

$2ap-2aq=q(p-1)d-p(q-1)d$

$2a(p-q)=d(qp-q-pq+p)$

$2a(p-q)=d(p-q)$

$2a=d$

So,

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

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

$\frac{a_p}{a_q}=\frac{1+(p-1)2}{1+(q-1)2}$

$\frac{a_p}{a_q}=\frac{1+2p-2}{1+2q-2}$

$\frac{a_p}{a_q}=\frac{2p-1}{2q-1}$

Therefore, Ratio of pth term and qth term is 2p - 1 : 2q - 1.