InterviewSolution
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InterviewSolution
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Let `(1+x^(2))^(2) (1+x)^(n) = sum_(k=0)^(n+4) a_(k)x^(k)`.. If `a_(1), a_(2)` and `a_(3)` aer in arithmetic progression, then the possible value/values of n is/areA. 5B. 4C. 3D. 2 |
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Answer» Correct Answer - B::C::D `L.H.S. = (1+2x^(2) + x^(4)) (1+ C_(1)x + C_(2)x^(2) + C_(3)x^(3) + "…..")` `R.H.S. = a_(0) + a_(1)x + a_(2)x^(2) + a_(3)x^(3) + "….."` Comparing the coefficients of `x, x^(2), x^(3),"….."` `a_(1) = C_(1), a_(2) = C_(2) + 2, a_(3) = C_(3) + 2C_(1) " "(1)` Now, `2a_(2) = a_(1) + a_(3) (A.P.)` `rArr 2(.^(n)C_(2) +2) = .^(n)C_(1) + (.^(n)C_(3) + 2.^(n)C_(1))` [Using (1)] or `2 (n(n-1))/(2) + 4 = 3n + (n(n-1)(n-2))/(6)` or `n^(3) -9n^(2) + 26n - 24 = 0` or `(n-2)(n^(2) - 7n + 12) = 0` or `(n-2)(n-3)(n-4) = 0` or `n = 2,3,4` |
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