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Question
Find the value of `(2 + sqrt(5))^5 + (2 - sqrt(5))^5`
Solution
`(2 + sqrt(5))^5 = ""^5"C"_0(2)^5 + ""^5"C"_1(2)^4 (sqrt(5)) + ""^5"C"_2 (2)^3 (sqrt(5))^2 + ""^5"C"_3 (2)^2 (sqrt(5))^3 + ""^5"C"_4 (2)(sqrt(5))^4 + ""^5"C"_5(sqrt(5))^5` ...(1)
and `(2 - sqrt(5))^5 = ""^5"C"_0 (2)^5 - ""^5"C"_1 (2)^4 (sqrt(5)) + ""^5"C"_2 (2)^3 (sqrt(5))^2 - ""^5"C"_3(2)^2(sqrt(5))^3 + ""^5"C"_4(2)(sqrt(5))^4 - ""^5"C"_5(sqrt(5))^5` ...(2)
Adding (1) and (2), we get,
`(2 + sqrt(5))^5 + (2 - sqrt(5))^5 = 2[""^5"C"_0(2)^5 + ""^5"C"_2(2)^3 (sqrt(5))^2 + ""^5"C"_4(2)(sqrt(5))^4]`
Now 5C0 = 1, 5C4 = 5C1 = 5, 5C2 = `(5 xx 4)/(1 xx 2)` = 10
∴ `(2 + sqrt(5))^5 + (2 - sqrt(5))^5` = 2[1 × 32 + 10 × 8 × 5 + 5 × 2 × 25]
= 2(32 + 400 + 250)
= 2(682)
= 1364
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