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Question
Prove that `(sqrt(5) + 1)^5 - (sqrt(5) - 1)^5` = 352
Solution
`(sqrt(5) + 1)^5 = ""^5"C"_0 (sqrt(5))^5 (1)^0 + ""^5"C"_1 (sqrt(5))^4 (1)^1 + ""^5"C"_2 (sqrt(5))^3 (1)^2 + ""^5"C"_3 (sqrt(5))^2 (1)^3 + ""^5"C"_4 (sqrt(5)^1) (1)^4 + ""^5"C"_5(sqrt5)^0 (1)^5`
Now, 5C0 = 5C5 = 1, 5C1 = 5C4 = 5, 5C2 = 5C3 = 10
∴ `(sqrt(5) + 1)^5 = 1(25sqrt5)(1) + 10(5sqrt5)(1) + 10(5)(1) + 5sqrt5(1) + 1(1)(1)`
∴ `(sqrt(5) + 1)^5 = 25sqrt5 + 125 + 50sqrt5 + 50 + 5sqrt5 + 1` ...(1)
Also, `(sqrt(5) - 1)^5 = ""^5"C"_0(sqrt(5))^5(1)^0 - ""^5"C"_1(sqrt(5))^4(1)^1 + ""^5"C"_2 (sqrt(5))^3(1)^2 - ""^5"C"_3 (sqrt(5))^2(1)^3 + ""^5"C"_4 (sqrt(5))^1(1)^4 - ""^5"C"_5(sqrt5)^0 (1)^5`
∴ `(sqrt(5) - 1)^5 = 1(25sqrt5)(1) - 5(25)(1) + 10(5sqrt5)(1) - 10(5)(1) + 5sqrt5(1) - 1(1)(1)`
∴ `(sqrt(5) - 1)^5 = 25sqrt5 - 125 + 50sqrt5 - 50 + 5sqrt5 - 1` ...(2)
Subtracting (2) from (1), we get,
`(sqrt(5) + 1)^5 - (sqrt(5) - 1)^5 = (25sqrt5 + 125 + 50sqrt5 + 50 + 5sqrt5 + 1) - (25sqrt5 - 125 + 50sqrt5 - 50 + 5sqrt5 - 1)`
= `25sqrt5 + 125 + 50sqrt5 + 50 + 5sqrt5 + 1 - 25sqrt5 + 125 - 50sqrt5 + 50 - 5sqrt5 + 1`
= `cancel(25sqrt5) + 125 + cancel(50sqrt5) + 50 + cancel(5sqrt5) + 1 - cancel(25sqrt5) + 125 - cancel(50sqrt5) + 50 - cancel(5sqrt5) + 1`
= 125 + 125 + 50 + 50 + 1 + 1
= 250 + 100 + 2
= 352
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