Induction

Induction - Preliminary to Math Induction An Infinite...

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Preliminary to Math Induction - An Infinite Sequence of Propositions: In Section 11.1, formulas are used to define an infinite sequence of numbers. For instance the sequence {1, 2, 4, 8, 16, ... } is generated by the formula 2 1 1 n n - {} = and { 1, 3, 5, 7, 9, ... } is generated by the formula 21 1 n n - = . In Section 11.4, math induction is used to prove all the propositions (or statements) in an infinite sequence of propositions are true. The sequence of propostions can by defined by a formula analogous to the formulas used to generate sequences of numbers. Notice the pattern in the following sequence of equations (propositions): 1 = 1 2 1 + 3 = 2 2 1 + 3 + 5 = 3 2 1 + 3 + 5 + 7 = 4 2 1 + 3 + 5 + 7 + 9 = 5 2 , etc. The n th equation in this sequence of equations is defined by: 1 + 3 + ... + (2n-1) = n 2 (Notice that in the nth equation: the number of terms in the sum on the left hand side is n, the last term in the sum on the left is given by the formula 2n-1 and the sum of the n terms equals n 2 .) Another sequence of propositions is: 1 = 1 * 2/ 2 1 + 2 = 2 * 3/ 2 1 + 2 + 3 = 3 * 4/ 2 1 + 2 + 3 + 4 = 4 * 5/ 2 1 + 2 + 3 + 4 + 5 = 5 * 6/ 2 , etc. This sequence of equations is defined by the formula: 1 + 2 + 3 + ... + n = n(n+1)/ 2 .
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Notation: P n will be used to denote the n th proposition in a sequence of propositions. If P n is defined by the formula 1 + 3 + ... + (2n-1) = n 2 , then the sequence of propositions which it generates is: P 1 : 1 = 1 2 P 2 : 1 + 3 = 2 2 P 3 : 1 + 3 + 5 = 3 2 P 4 : 1 + 3 + 5 + 7 = 4 2 P 5 : 1 + 3 + 5 + 7 + 9 = 5 2 , etc. In proof by induction, we must show that IF a PARTICULAR proposition in a sequence of propositions is TRUE, then the NEXT proposition in the sequence must ALSO be TRUE. (Recall the recursive definition of a sequence. If we are able to calculate a particular term in a recursive sequence, then we could use that value to calculate the next term in the sequence. Just as a recursive definition has two parts - the part which defines the initial term(s) in the sequence and the part which gives the rule used to generate subsequent terms in the sequence, so a proof by math induction has two parts - the Basis Step (which establishes the truth of the sequence of propositions for an initial value(s)) and the Inductive Step (which establishes that IF a particular proposition in a sequence of propositions is TRUE, then the NEXT proposition in the sequence must ALSO be TRUE.)) The notation we will use for the PARTICULAR proposition is P k and for the NEXT proposition is P k+1 .
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This note was uploaded on 12/11/2011 for the course MAC 1140 taught by Professor Kutter during the Fall '11 term at FSU.

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Induction - Preliminary to Math Induction An Infinite...

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