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in Binomial Theorem, Sequences and Series by (46.9k points)
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If the ratio of the sums of m terms and n terms of an A.P. be m2 : n2, prove that the ratio of its mth and nth terms is (2m – 1) : (2n – 1).

1 Answer

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Best answer

Let ‘a’ be the first term and d be the common difference of A.P.

Using the given information, we have

⇒ 2an + (mn – n)d = 2am + (mn – m)d 

⇒ 2an – 2am = (mn – m – mn + n)d 

⇒ 2a(n – m) = (n – m)d ⇒ d = 2a, [n – m ≠ 0, as n ≠ m]

Now, 

Hence, the required ratio is (2m – 1) : (2n – 1)

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