mathematical induction3.3

(a) (1) S1 = 1
(2) S2 = 1 + 2 + 1 = 4
(3) S3 = 1 + 2 + 3 + 2 + 1 = 9
(b) Sn may be equal to n^2
Let P(n) be the proposition that Sn = n^2 for all positive integers n
P(1) is true as S1 = 1
Assume P(k) is true, i.e. Sk = 1 + 2 + 3 + ... + (k - 1) + k + (k - 1) + ... + 3 + 2 + 1 = k^2
Sk+1 = 1 + 2 + 3 + ... + (k - 1) + k + (k + 1) + k + (k - 1) + ... + 3 + 2 + 1
= Sk + 2k + 1
= k^2 + 2k + 1
= (k + 1)^2
=> P(k + 1) is also true.
Hence by the principle of MI, P(n) is true for all positive integers n.

1. S1 = 1

S2 = 1 + 2 + 1 = 4

S3 = 1 + 2 + 3 + 2 + 1 = 9

b. The formula of Sn = n2

2009-10-06 21:08:23 補充：
Let P(n) be the statement, that

Sn = n2, for all positive integers n

When n = 1, the answer is trivial.

P(1) is true.

Assume P(k) is true for n = k

Sk = k2

2009-10-06 21:08:27 補充：
When n = k + 1

Sk+1 = 1 + 2 + ... + k + (k + 1) + k + ... + 2 + 1

= k + (k + 1) + [1 + 2 + ... + (k - 1) + k + (k - 1) + ... + 2 + 1]

= 2k + 1 + Sk

= 2k + 1 + k2

= (k + 1)2

By the Principle of M.I., P(n) is true for all positive integers n.