What is Induction: Definition and 999 Discussions

Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, . . . ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), . . . . Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder:

Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step).
A proof by induction consists of two cases. The first, the base case (or basis), proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that if the statement holds for any given case n = k, then it must also hold for the next case n = k + 1. These two steps establish that the statement holds for every natural number n. The base case does not necessarily begin with n = 0, but often with n = 1, and possibly with any fixed natural number n = N, establishing the truth of the statement for all natural numbers n ≥ N.
The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction is an inference rule used in formal proofs, and in some form is the foundation of all correctness proofs for computer programs.Although its name may suggest otherwise, mathematical induction should not be confused with inductive reasoning as used in philosophy (see Problem of induction). The mathematical method examines infinitely many cases to prove a general statement, but does so by a finite chain of deductive reasoning involving the variable n, which can take infinitely many values.

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  1. A

    Different statement of strong induction proof

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  2. K

    MHB Proving the Recursion Formula for $(x_n)$ Using Strong Induction

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  3. F

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  4. N

    MHB Solve Induction Problem: cn <= n log2n

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  5. S

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  6. R

    What is the inductive step for proving mathematical induction?

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  7. M

    Engineering Speed control of induction motor in steady state

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  8. MarkFL

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  9. Omega0

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  10. S

    Explaining the Inconsistency and Solution in Faraday's Law of Induction

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  11. W

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  12. L

    Prove by Induction: Un = 1/2(1 - 1/22^n)

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  13. M

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  14. S

    Proving an Inequality by Induction

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  15. srfriggen

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  16. M

    Engineering Inverter - V/F constant control of an induction motor

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  17. M

    Engineering V / F constant control of induction motor with inverter

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  18. S

    Polarization of Electromagnetic Wave and Faraday's Law of Induction

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  19. K

    MHB Induction for writing integers

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  20. K

    MHB Induction for divisibility by 10

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  21. K

    MHB Induction for series of squares

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  22. R

    Some Qustions of Electrostatic Induction

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  23. K

    MHB Prove 1/√n Inequality by Induction

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  24. P

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  25. T

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  26. Q

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  27. H

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  28. S

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  29. M

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  30. skate_nerd

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  31. N

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  32. srfriggen

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  33. P

    Show by induction that (2n-1) /(2n) <= 1/(3n+1) for all n

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  34. K

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  35. C

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  36. S

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  37. C

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  38. F

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  39. B

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  40. J

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  41. V

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  42. L

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  43. A

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  44. J

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  45. G

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  46. S

    MHB Can we prove $3^n > n^3$ for all $n \geqslant 4$ using induction?

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  47. P

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  48. C

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  49. R

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  50. G

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