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Question: The value of \((\sqrt{2} + 1)^{6} + (\sqrt{2} - 1)^{6}\) will be...

The value of (2+1)6+(2−1)6(\sqrt{2} + 1)^{6} + (\sqrt{2} - 1)^{6} will be

A

– 198

B

198

C

98

D

– 99

Answer

198

Explanation

Solution

We know that,

(x+y)n+(x−y)n=2[xn+n⥂C2xn−2y2+n⥂C4xn−4y4+.....](2+1)6+(2−1)6=2[(2)6+6⥂C2(2)4(1)2+6⥂C4(2)2(1)4+6⥂C6(2)0(1)6](x + y)^{n} + (x - y)^{n} = 2\lbrack x^{n} +^{n} ⥂ C_{2}x^{n - 2}y^{2} +^{n} ⥂ C_{4}x^{n - 4}y^{4} + .....\rbrack(\sqrt{2} + 1)^{6} + (\sqrt{2} - 1)^{6} = 2\lbrack(\sqrt{2})^{6} +^{6} ⥂ C_{2}(\sqrt{2})^{4}(1)^{2} +^{6} ⥂ C_{4}(\sqrt{2})^{2}(1)^{4} +^{6} ⥂ C_{6}(\sqrt{2})^{0}(1)^{6}\rbrack = 2[8+15×4+30+1]=1982\lbrack 8 + 15 \times 4 + 30 + 1\rbrack = 198