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प्रश्न
Write the nth term of the sequence `3/(1^2 2^2), 5/(2^2 3^2), 7/(3^2 4^2), ...` as a difference of two terms
उत्तर
The given sequence is `3/(1^2 * 2^2), 5/(2^2 * 3^2), 7/(3^2 * 4^2), ....`
The terms in the numerator are 3, 5, 7
Which forms an A. P with first term a = 3 and common difference d = 5 – 3 = 2
nth term tn = a + (n – 1)d
= 3 + (n – 1)(2)
= 3 + 2n – 2
= 2n + 1
tn = 2n + 1
The terms in the denominator are 12 . 22, 22 . 32, 32 . 42 ...
nth term tn = n2 . (n + 1)2
∴ nth term of the given sequence is an = `(2"n" + 1)/("n"^2 ("n" + 1)^2)`
= `("n"^2 + 2"n" + 1 - "n"^2)/("n"^2 ("n" + 1)^2)`
= `(("n" + 1)^2 - "n"^2)/("n"^2 ("n" + 1)^2)`
= `("n" +1)^2/("n"^2("n" + 1)^2) - "n"^2/("n"^2 ("n" + 1)^2`
an = `1/"n"^2 - 1/("n" + 1)^2`
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