Mathematical induction
1. What is mathematical induction
Mathematical induction is a proven method for existing conclusions (such as guessing what conclusions) of.
II. How mathematical induction
1. First induction
For any natural numbers proposition \ (P (n-) \) , if the \ (P (0) = to true \;, \; P (n-) \ Rightarrow P (n-+. 1) \) , the proposition for all natural number set up.
Example: Show that
\ [\ Sigma_ {i = 1
} ^ {n} \; i ^ 3 = [\ frac {n (n + 1)} {2}] ^ 2 \] Proof: Clearly, for i = 1, the equation is established.
\ [\ Begin {align} & For k> 1 \\ & [\ frac {(k + 1) (k + 2)} {2}] ^ 2 - [\ frac {k (k + 1)} {2 }] ^ 2 = (k + 1) ^ 3 \\ & \ because P (1) = true \\ & \ therefore established \ end {align} \]
2. The second induction
For arbitrary propositional natural numbers P (n), if on \ (\ bigcap_ {K =. 1} ^ {K <n-} P (K) \; \ Rightarrow P (n-) \) , the proposition for any non- founded negative integer n
Example: For a positive integer n (n> 1), it can be divided into a + b (1 <= a, b, a, b are integers), then the resulting value is a * b..
Prove:
\ [n integer points make the resulting score is \ frac {(n-1)
n} {2} \] Syndrome:
\ [\} the begin {align = left & \ Because \ bigcap_. 1 = {I } ^ {n-1} P (i) = \ frac {(i-1) i} {2} \\ \ therefore P (n) & = P (a) + P (b) + a * b \\ & = \ frac {(a- 1) a + (b-1) b + 2ab} {2} \\ & = \ frac {(a + b) ^ 2- (a + b)} {2} \\ & = \ frac {(n + 1
) (n)} {2} \ end {align} \] it is clear that P (2) compliance, proof.