## Mathematical Induction

Mathematical
Induction

Mathematical induction is
a form of mathematical proof.

Just because a rule, pattern, or formula seems to

work for several values of n, you cannot simply

decide that it is valid for all values of n without

going through a legitimate proof.

The Principle of
Mathematical Induction

Let Pn be a statement
involving the positive

integer n. If

1.P1 is true, and

2.the truth of Pk
implies the truth of Pk+1 , for

every
positive integer k,

then Pn must be true for all
integers n.

Ex. Use mathematical
induction to prove the following

formula.

Sn = 1 + 3 + 5 + 7 + . . . + (2n-1) = n2

First, we must show that the formula works for n = 1.

1.For n = 1

S1 = 1 = 12

The second
part of mathematical induction has two

steps. The first step is to assume that the formula
is

valid for
some integer k. The second step is to
use this

assumption
to prove that the formula is valid for
the

next
integer, k + 1.

2.Assume Sk = 1
+ 3 + 5 + 7 + . . . + (2k-1) = k2

is true, show that Sk+1 = (k + 1)2
is true.

Sk+1 = 1 + 3 + 5 + 7 + . . . + (2k – 1) +
[2(k + 1) – 1]

= [1 + 3 + 5 + 7 + . . . +(2k – 1)] +
(2k + 2 – 1)

= Sk + (2k + 1)

= k2 + 2k + 1

=(k + 1)2

**To read complete topic ...................click here**

**other Mathematics topics............**

**Series-formulas**

The-binomial-theorem

Analytic-trigonometry-double-angle-half angle

solved-problems-on-limits-and-continuity

geometric-progressions

Rationalizing-denominator-of-radicals

Basic-point-formulas-distance-midpoint

Two-point-form-normal-form parametric

limit-questions

set-relation-function-part1

permutation-combination

straight-lines

Three-diamensional-geometry

complex-numbers-and-quadratic-equations

Trigonometry

hyperbola

### Inequalities: An Introduction

### Cartesian Coordinate Geometry and Straight Lines

### Introduction of Conic section ,circle ,ellipse

### Rationalizing the Denominator of Radicals Expressions

**Download supplementary material**Download

## 0 comments:

## Post a comment