j The statement remains the same: S This could be called "predecessor induction" because each step proves something about a number from something about that number's predecessor. {\displaystyle A} = mathematical proof typically used to establish a given statement for all natural numbers Consider the statement that "every natural number greater than 1 is a product of (one or more) prime numbers", which is the "existence" part of the fundamental theorem of arithmetic. 2 2 | To view this arrangement more clearly, we’ll rewrite both series using summation notation. . + ) F {\displaystyle n\geq 3} k , could be proven without induction; but the case {\displaystyle j} Indeed, suppose the following: It can then be proved that induction, given the above-listed axioms, implies the well-ordering principle. {\displaystyle P(k)\implies P(k{+}1)} {\displaystyle 0+1+\cdots +k\ =\ {\frac {k(k{+}1)}{2}}.}. and At first glance, it may appear that a more general version, ) ( + Therefore, by the complete induction principle, P(n) holds for all natural numbers n; so S is empty, a contradiction. k The reader may find a video walk-through of Pascal’s Rule helpful. {\displaystyle n_{1}} x 1 This form of induction has been used, analogously, to study log-time parallel computation. ⟹ {\displaystyle k} + Induction is often used to prove inequalities. {\displaystyle 0+1={\tfrac {(1)(1+1)}{2}}} {\displaystyle n=1} 1 2 12 ( Although the form just described requires one to prove the base case, this is unnecessary if one can prove P(m) (assuming P(n) for all lower n) for all m ≥ 0. Induction hypothesis: Given some 0 1 prove by induction ∑k = 1n k ( k + 1) = n ( n + 1) ( n + 2) 3. induction-calculator. The result of all that effort is Equation 15. ⋯ The Binomial Theorem tells us how to expand a binomial raised to some non-negative integer power. {\displaystyle F_{n+2}=F_{n+1}+F_{n}} 0 In this form the base case is subsumed by the case m = 0, where P(0) is proved with no other P(n) assumed; 12 Mathematical Induction. {\displaystyle P(k{+}1)} {\displaystyle m} {\displaystyle m} if one assumes that it already holds for both More complicated arguments involving three or more counters are also possible. {\displaystyle m} {\displaystyle k\in \{4,5,8,9,10\}} + k For example, Augustin Louis Cauchy first used forward (regular) induction to prove the ) k F n {\displaystyle S(j-4)} As an example, we prove that If traditional predecessor induction is interpreted computationally as an n-step loop, then prefix induction would correspond to a log-n-step loop. ) ( also holds for is a variable for predicates involving one natural number and k and n are variables for natural numbers. ⋯ n Using mathematical induction on the statement P(n) defined as "Q(m) is false for all natural numbers m less than or equal to n", it follows that P(n) holds for all n, which means that Q(n) is false for every natural number n. The most common form of proof by mathematical induction requires proving in the inductive step that. for all natural numbers . Because of that, proofs using prefix induction are "more feasibly constructive" than proofs using predecessor induction. = x {\displaystyle k=12} 0 , , Finally, we reintegrate the last term into the summation by changing t to t+1 (above the sigma). Sometimes, it is more convenient to deduce backwards, proving the statement for {\displaystyle n} ≤ | ∎. n The earliest rigorous use of induction was by Gersonides (1288–1344). + the statement holds for all smaller ) ≥ 12 n The name "strong induction" does not mean that this method can prove more than "weak induction", but merely refers to the stronger hypothesis used in the inductive step. = + Proposition. ∈ ≥ {\displaystyle 0={\tfrac {0(0+1)}{2}}\,.}. n For any N holds for all natural numbers j {\displaystyle S(j)} Any set of cardinal numbers is well-founded, which includes the set of natural numbers. = F | 4 ) 1 as follows: Base case: Showing that k Take a look, Survivorship Bias: The Mathematician Who Helped Win WWII, 25 Interesting Books for Math People and Designers. Suppose there is a proof of P(n) by complete induction. {\displaystyle 0} {\displaystyle n\geq -5} P 1 + The proof consists of two steps: The hypothesis in the inductive step, that the statement holds for a particular 2 ( ) 1 1 + ( 5 15 b F The first term of Series A juts out to the left. Proof. , It is mistakenly printed in several books and sources that the well-ordering principle is equivalent to the induction axiom. = We then follow that assumption to its logical conclusion. Proof by mathematical induction. n {\displaystyle P(n)} {\displaystyle F_{n}} {\displaystyle m=10} + x 2 Then the base case P(0,0) is trivially true, and so is the step case: if P(x,n), then P(succ(x,n)). We assume that the theorem is true for some integer, t. We show that if the theorem applies to some integer t, it must also apply to the integer t+1. − n 1 Mathematical induction is an inference rule used in formal proofs, and in some form is the foundation of all correctness proofs for computer programs. ( 4 We shall look to prove the same example as above, this time with strong induction. m S

## proof by mathematical induction

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