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Integration to solve modified sum of binomial coefficients

The value of ...

Question

The value of $(1 + \sqrt{2})^{7} + (1 - \sqrt{2})^{7}$ is

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Solution

We know that,
$(1 + x)^{n} + (1 - x)^{n} = 2 (^{n} C_{0} +^{n} C_{2} x^{2} +^{n} C_{4} x^{4} + \dots)$

So, $(1 + \sqrt{2})^{7} + (1 - \sqrt{2})^{7}$
$= 2 (^{7} C_{0} +^{7} C_{2} (\sqrt{2})^{2} +^{7} C_{4} (\sqrt{2})^{4} +^{7} C_{6} (\sqrt{2})^{6}) = 2 (1 + 42 + 140 + 56) = 478$

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Integration to solve modified sum of binomial coefficients

Standard XI Mathematics

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