Mathematical Indunction

Ch. 2 Mathematical Indunction Exerxise Solutions

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 Induction  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 Indunction, 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 = Mathematical Induction 20=Mathematical Induction 21  = 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  M.I.  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  M.I. 

   P(n) is true for all n ∈ N

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

3 . Mathematical Induction 12

Sol:

let P(n) be the given statement that

Mathematical Induction 12

       For n = 1

        LHS = Mathematical Induction 13 =Mathematical Induction 14

        RHS =Mathematical Induction 15  =Mathematical Induction 16  = Mathematical Induction 14

       LHS = RHS

       P (n) is true for n = 1

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

Mathematical Induction 17  ………… (1)

For n = k + 1

Add Mathematical Induction 22on both sides of (1)

Mathematical Induction 18 

                                                                                                                             Mathematical Induction 11

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

∴ By the principle of M.I.   P(n) is true for all n ∈ N

∴   Mathematical Induction 12

 

4. 43 + 83 + 123 + … up to n terms = 16 n2 (n + 1)2

Sol:

let P(n) be the given statement that

 4, 8, 12, … are in AP

a = 4, d = 4

an = a + (n – 1) d

     = 4 + (n – 1)4

     = 4 + 4n – 4

      = 4n

nth term of given series is (4n)3

let P(n) be the given statement that

43 + 83 + 123 + … + (4n)3= 16 n2 (n + 1)2

       For n = 1

        LHS = 43 = 63

        RHS = 16 (1)2 (1 + 1)2 = 16 × 4 = 64

       LHS = RHS

       P (n) is true for n = 1

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

43 + 83 + 123 + … + (4k)3= 16 k2 (k + 1)2 ………… (1)

      For n = k + 1

     Add [4 (k + 1)]3 on both sides of (1)

    43 + 83 + 123 + … + (4k)3= 16 k2 (k + 1)2 + [4 (k + 1)]3

                                                      = 16 k2 (k + 1)2 + 64 (k + 1)3

                                                      = 16 (k + 1)2 [k2 + 4 (k + 1)]

                                                      = 16 (k + 1)2 [k2 + 4 k + 4]

                                                      = 16 (k + 1)2 (k + 2)2

                                                      = 16 (k + 1)2 ( Mathematical Induction 23+ 1)2

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

∴ By the principle of Mathematical induction

   P(n) is true for all n ∈ N

5. a + (a + d) + (a + 2d) + …. up to n terms = Mathematical Induction 24

Sol:

Given series is a + (a + d) + (a + 2d) + …. up to n terms = Mathematical Induction 24

nth term of the given series is a + (n – 1) d

     let P(n) be the given statement that

      a + (a + d) + (a + 2d) + …. +[a + (n – 1) d] = Mathematical Induction 24

     for n = 1

     LHS = a;     RHS = Mathematical Induction 25 = a

     LHS = RHS

     P(n) is true for n = 1

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

   a + (a + d) + (a + 2d) + …. + [a + (k – 1) d] = Mathematical Induction 26……. (1)

   add (a + k d) on both sides of (1)

    a + (a + d) + (a + 2d) + …. [a + (k – 1) d] + [a + kd] = Mathematical Induction 26 + [a + kd]

 

             Mathematical Induction 28

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

∴ By the principle of Mathematical induction

   P(n) is true for all n ∈ N

Mathematical Induction

 


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