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` are in arithmetic progression, then the possible value/values of `n` is/are a. 5 b. 4 c. `3` d. `2`

Options

A. 5
B. 4
C. 3
D. 2

Answer

B, C, D

Explanation

`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`