請教高一翰林數學問題3題?

1.
2/[(1/a) + (1/b)] = 2ab/(a+b) = 4ab/2(a+b)
(a+b)/2 = (a+b)^2/2(a+b)
[(a+b)^2/2(a+b)] - [4ab/2(a+b)] = (a-b)^2/2(a+b) >= 0
所以(a+b)/2 > 2/[(1/a) + (1/b)]
當a=b 時,則兩者相等…………..(解答)

2.
x = √(5+2√6)= √(√3 + √2)^2 = √3 + √2
1/x = 1/(√3 + √2) = √3 - √2
x+(1/x) = 2√3

x^2 + (1/x^2) = (√3 + √2)^2 + (√3 - √2)^2 = 10

x^3 + (1/x^3)
= [x+(1/x)]*[ x^2 + (1/x^2)] – [x+(1/x)]
= 2√3 * 10 - 2√3
= 18√3…………..(解答)

3.
√2 = 1 + 1/(2+x)
1/(2+x) = √2 – 1
2+x = 1/(√2 – 1) = √2 + 1
x = √2 – 1

√2 = 1 + 1/[2 + (1/(2+x))]
1/[2 + (1/(2+x))] = √2 – 1
2 + (1/(2+x)) = 1/(√2 – 1) = √2 + 1
1/(2+x) = √2 – 1
2+x = 1/(√2 – 1) = √2 + 1
x = √2 – 1……………(解答)

希望有幫上你的忙!