高2推理与证明数学题

来源：百度知道 编辑：UC知道 时间：2024/06/13 21:35:04
已知数列{an}的前n项和为Sn,a1=-2/3,满足Sn+1/Sn+2=an(n>=2),计算S1,S2,S3,S4,并猜想Sn表达式.

S(1) = a(1) = -2/3

S(2) = S(1) + a(2) = a(2) - 2/3, a(2) = S(2) + 2/3

S(2) + 1/S(2) + 2 = a(2) = S(2) + 2/3,

1/S(2) = -4/3,

S(2) = -3/4

S(3) = S(2) + a(3) = a(3) - 3/4, a(3) = S(3) + 3/4

S(3) + 1/S(3) + 2 = a(3) = S(3) + 3/4,

1/S(3) = -5/4,

S(3) = -4/5.

S(4) = S(3) + a(4) = a(4) - 4/5, a(4) = S(4) + 4/5.

S(4) + 1/S(4) + 2 = a(4) = S(4) + 4/5,

1/S(4) = -6/5,

S(4) = -5/6.

猜想
S(n) = -(n+1)/(n+2), n = 1,2,...

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证明
n = 1时，S(1) = -2/3,结论成立。
设 n = k时，有S(k) = -(k+1)/(k+2)

当n = k+1时，
S(k+1) = S(k) + a(k+1) = a(k+1) - (k+1)/(k+2),
a(k+1) = S(k+1) + (k+1)/(k+2),

S(k+1) + 1/S(k+1) + 2 = a(k+1) = S(k+1) + (k+1)/(k+2),

1/S(k+1) = -(k+3)/(k+2),

S(k+1) = -(k+2)/(k+3),结论成立。

所以，
由归纳法，猜想正确。

楼上的方法很好嘛！归纳法虽说是大学的课程，但是判高考卷的都是大学老师，所以嘛。。。什么方法都行！