MATHEMATICAL INDUCTION

In this section, we look at a form of proof, the principle of mathematical induction consider the pattern below

From the pattern formed above by the first five sums, it appears that the sum of first n is

When this formula is correct, it is vital for you to see that recognizing and arriving at the conclusion that the pattern is true for all values of n is not logically valid of proof

One of such logical proof we use to validity of a formula is the principle of mathematical induction

 

PRINCIPLE OF MATHEMATICAL INDUCTION

This is used to prove the validity of proposition of the set of non – negative integers. To prove by induction, first we show that the proposition is valid for the least element of N. We then assume that the proposition is true for an arbitrary element of N say K where K is an element; we then show that proposition is valid for the next value after K i.e.

K + 1

Summary

Let P_n be a statement involving positive integer n if the two condition are satisfied

1.                    P₁ is true (i.e. P_n  is true when n = 1)

2.                    If P_n is true when n = k then P_n is also true when K + 1 then P_n must be true for all integers

Example1

Prove by induction

Solution

 

Example 2

Use the principle mathematical induction to prove that

Solution

 

Example 3

Use mathematical induction to establish the

Solution 

 

 

Example 4

Prove by method of mathematical induction that

 

Solution

 

Example 5

Use mathematical induction to prove

Solution

 

Example 6

Using mathematical induction prove

 

Solution

When n = 1

L.H.S    4(1) – 1 =3

R.H.S    (1)[2(1) + 1}=3

Thus the statement is valid when n =1

Assuming that the proposition is true for n = k

Example 7

Use mathematical induction to prove the given formula for every positive integer n

Solution

SUM OF POWERS OF INTEGERS

Below is a list of collection of useful summation formula dealing with the sum of various powers of the first n positive integers

Example 8

Solution

Using the formula for the sum of  of the first n positive integers

Example

Find the indicated sum using formula for the sum of powers of integers

Solution