The first mathematical induction
The first mathematical induction can be summarized as the following three steps:
- (1) induction foundation: proof n = 1 Proposition established;
- (2) induction hypothesis: Suppose n = k proposition is established when;
- (3) induction recursion: By induction hypothesis Release n = k + 1 Proposition also set up.
- Thus can be concluded from the propositions are true for all positive integers.
The second mathematical induction (full induction)
The second principle of mathematical induction is a positive integer n has about the proposition, if:
- (1) induction foundation: when n = 1,2 , the proposition is true;
- (2) induction hypothesis: suppose that when n≤k time (k∈N), the proposition was established;
- (3) induction recursion: whereby appropriate push n = k + 1 , the proposition is also true.
- So according to ①② available, propositions are true for all n is a positive integer.
example
Monotone bounded guidelines, the number of columns
Provided. 1 = A1, \ (A_. 1} + {√ n-+ (. 1-AN) = 0 \) , {an} convergence proof, and determining the \ (n-lim_ → ∞} A_N {\) .
- If the limit is set to A exist, + √ A (. 1-A) = 0, A = (-. 1-√5) / 2
A1 =. 1, A2 = 0, A3 = -1, so guess {an} monotonically decreasing , there is a lower bound - Following by a second mathematical induction to prove {an} :( generally monotonically decreasing for monotonicity )
- n = 1, n when 2 =, a1 = 1, a2 = 0, a1> a2
- Assuming n≤k, \ (K-A_. 1 {}> {K} A_ \) established
- When K +. 1 = n-, \ (K + A_. 1} = {- √ (. 1-a_k) <- √ (. 1. 1-K-A_ {}) = a_k established \)
- Therefore, {an} monotonically decreasing
- Below with a first mathematical induction to prove a lower bound :( {an} has a generally vertical boundary )
- n = 1, a1 = 1> (- 1-√5) / 2 established
- Assuming that n = k, ak> (- 1-√5) / 2 established
- 当n = k + 1时, \ (a_ {k + 1} = - √ (1-a_k) \) > (- 1-√5) / 2
1-if <(3 + √5) / 2 = ( 1 + 5 + 2√5) / 4
√ (1-k) <(1 + √5) / 2