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Question: The first 3 terms in the expansion of \((1 + x - 3x^{2})^{2134}\) \(2^{2134}\) are 1, 6x and 16x<sup...

The first 3 terms in the expansion of (1+x−3x2)2134(1 + x - 3x^{2})^{2134} 221342^{2134} are 1, 6x and 16x2. Then the value of a and n are respectively.

A

2 and 9

B

3 and 2

C

2/3 and 9

D

3/2 and 4n4^{n}

Answer

2/3 and 9

Explanation

Solution

f=f′f = f^{'} …..(i)

∴Rf=Rf′=(55+11)2n+1(55−11)2n+1\therefore Rf = Rf^{'} = (5\sqrt{5} + 11)^{2n + 1}(5\sqrt{5} - 11)^{2n + 1} …..(ii)

[(55)2−112]2n+1=42n+1\lbrack(5\sqrt{5})^{2} - 11^{2}\rbrack^{2n + 1} = 4^{2n + 1} …..(iii)

From (ii), (2+1)6=k+f,(\sqrt{2} + 1)^{6} = k + f, ⇒ (0≤f<1)(0 \leq f < 1) …..(iv)

From (iii), (2−1)6=f′,(0<f′<1)(\sqrt{2} - 1)^{6} = f',(0 < f' < 1) …..(v)

Only (3) is satisfying equation (iv) and (v).