Question

Find the value of
$(2+\sqrt{5})^{5}+(2-\sqrt{5})^{5}$

Answer 1

1364 is the required answer

Answer 2

Solution :

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By Binomial theorem :

( x + a )ⁿ = ⁿC0 xⁿ+ⁿC1 x^n-1 a + ⁿC2 x^n-2a²

+ ..+ ⁿCr x^n-r a^r + ... ⁿCn a^r

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$(2+\sqrt{5})^{5}+(2-\sqrt{5})^{5}$

= 2[ C0 2^5 +C2 2³(√5)² +C4 2×(√5)⁴ ]

= 2[ 32 + 10 × 8 × 5 + 5 × 2 × 25]

= 2[ 32 + 400 + 250 ]

= 2 × 682

= 1364

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