Math, asked by pratikmagadum7332, 11 months ago

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.

Answers

Answered by kritarth24
1

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Answered by aquialaska
3

Answer:

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

Step-by-step explanation:

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

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

We know Formula of sum of terms and a term of AP,

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

a_n=a+(n-1)d

Consider given statement,

\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

Now,

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

         =\frac{a+(p-1)2a}{a+(q-1)2a}

         =\frac{1+(p-1)2}{1+(q-1)2}

         =\frac{1+2p-2}{1+2q-2}

         =\frac{2p-1}{2q-1}

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

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