Mathematical Indunction (M.I) Exerxise wise Solutions

Mathematical Indunction

Mathematical Indunction (M.I) Exerxise wise Solutions

Mathematical Indunction: It is a technique for proving results or establishing statements for natural numbers. This part illustrates the method through a variety of examples.

Definition:
Mathematical Indunction  is a mathematical technique which is used to prove a statement, a formula or a theorem is true for every natural number.

The technique involves steps to prove a statement, as stated below −
Let P(n) or S(n) be the given statement

Step 1: For n =1
                 we get LHS = RHS
                 then P(n) is true for n= 1
Step 2: Let us assume thet P(n) is true for n =k
Step 3: We have to Prove P(n) is true for n= k + 1

Laplace:

Laplace was a mathematecian and astronomer whose work was pivotal to the development of mathematical astronomy. His most outstanding work was done in the fields of celestial mechonics, probability, differential equations, and geodesy. His five volume work on celestial mechonics earned him the title of the Newton of France.

Laplace

“Analysis and natural philosophy owe their most important discoveries to this fruitful means, which is called indunction” – Pierr Simon de Laplace


Exercise 2(a)

Using Mathematical Induction, Prove each of the following statement for all n ∈ N.

1. 12 + 22 + 32 + …… + n2= Mathematical Induction 1

Let p(n) be the given statement that

 12 + 22 + 32 + …… + n2=Mathematical Induction 1

For n= 1

LHS = 12 = 1

RHS =  =  = 1

LHS = RHS

P(n) is true for n = 1

Let us assume that P(n) is true for n = k

i.e., 12 + 22 + 32 + …… + k2=  ………… (1)

for n = k + 1

add (k +1)2 on both sides of (1)

12 + 22 + 32 + …… + k2 + (k +1)2 =Mathematical Induction 4 

                                                              Mathematical Induction 5   

                                                             Mathematical Induction 6

P(n) is true for n = k+ 1

∴ By the principle of Mathematical induction

   P(n) is true for all n ∈ N

∴ 12 + 22 + 32 + …… + n2=Mathematical Induction 1

2.  2.3 + 3.4 + 4.5 + ……… up to n terms = Mathematical Induction 7

First factors of given series are: 2, 3, 4, 5, …

                                  a = 2, d = 1

                                 an = a + (n – 1) d

                                       = 2 + (n – 1) (1)

                                       = 2 + n – 1

                                       = n + 1

Second factors of given series are: 3, 4, 5,…

                                  a = 3, d = 1

                                 an = 3 + (n – 1) d

                                       = 3 + (n – 1) (1)

                                       = 3 + n – 1

                                       = n + 2

nth term of given series is (n + 1) (n + 2)

let P(n) be the given statement that

2.3 + 3.4 + 4.5 + ……… + (n + 1) (n + 2) = Mathematical Induction 7

For n = 1

LHS = 2.3 = 6

RHS =  =  =  = 6

LHS = RHS

P(n) is true for n = 1

Let us assume that P(n) is true for n = k

i.e., 2.3 + 3.4 + 4.5 + ……… + (k + 1) (k + 2) = Mathematical Induction 8 ………… (1)

for n = k + 1

  add (k + 2) (k + 3) on both sides of (1)

 2.3 + 3.4 + 4.5 + ……… + (k + 1) (k + 2) + (k + 2) (k + 3)  

  =  Mathematical Induction 8 + (k + 2) (k + 3)

Mathematical Induction 9

Mathematical Induction 10

Mathematical Induction 11

P(n) is true for n = k+ 1

∴ By the principle of Mathematical induction

   P(n) is true for all n ∈ N

∴ 2.3 + 3.4 + 4.5 + ……… up to n terms =Mathematical Induction 7

 

 

 

 

 

 


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