Question

数列我都做疯了，请高手帮忙

Answer 1

(1)
令b(n) = 1 / S(n), S(n) = 1 / b(n)
当n>1时
S(n) - S(n-1) = a(n) = 2S(n)^2 / ( 2S(n) - 1 )
1/b(n) - 1/b(n-1) = 2 / b(n)^2 / ( 2 / b(n) - 1 )
整理得b(n) - b(n-1) = 2
{b(n)}等差, 所以{1/S(n)}为等差数列.

(2)
1/S(n) - 1/S(n-1) = 2
所以 1/S(n) = 1 + 1/S(1) * 2 * (n-1)
1/S(1) = 1/a(1) = 1
S(n) = 1 / ( 2n - 1 )
a(n) = S(n) - S(n-1) = 1 / ( 2n - 1 ) - 1 / ( 2n - 3 ) = -2 / ( 4n^2 - 8n + 3 )