Question 26: is divisible by for all

Answer:

Let is divisible by for all

For , L.H.S which is divisible by

Hence is true for

Let is true for

is divisible by for all

… … … … … i)

Now we have to show that is true for

We know that is divisible by . Hence

is true for

Hence by the principle of mathematical induction

is divisible by for all

Question 27: is divisible by for all

Answer:

Let is divisible by for all

For , L.H.S which is divisible by

Hence is true for

Let is true for

is divisible by for all

… … … … … i)

Now we have to show that is true for

where

is true for

Hence by the principle of mathematical induction

is divisible by for all

Question 28: for all

Answer:

Let for all

For , L.H.S R.H.S Therefore LHS = RHS

Hence is true for

Let is true for

for all … … i)

Now we have to show that is true for

is true for

Hence by the principle of mathematical induction

for all

Question 29: is divisible by for all

Answer:

Let is divisible by for all

For , L.H.S which is divisible by

Hence is true for

Let is true for

is divisible by for all

… … … … … i)

Now we have to show that is true for

where

is true for

Hence by the principle of mathematical induction

is divisible by for all

Question 30: for all

Answer:

Let for all

For , L.H.S R.H.S Therefore LHS = RHS

Hence is true for

Let is true for

for all … … … … … i)

Now we have to show that is true for

is true for

Hence by the principle of mathematical induction

for all

Question 31:

Answer:

Let

For , L.H.S R.H.S Therefore LHS = RHS

Hence is true for

Let is true for

… … … … … i)

Now we have to show that is true for

is true for

Hence by the principle of mathematical induction

Question 32: Prove that is a positive integer got all

Answer:

Let is a positive integer got all

For , L.H.S which is positive.

Hence is true for

Let is true for

is a positive integer got all … … … … … i)

Now we have to show that is true for

is true for

Hence by the principle of mathematical induction

is a positive integer got all

Question 33: Prove that is a positive integer got all

Answer:

Let is a positive integer got all

For ,

L.H.S

1 is an integer. Hence is true for

Let is true for

is a positive integer got all

Now we have to show that is true for

Which is an integer. is true for

Hence by the principle of mathematical induction

is a positive integer got all

Question 34: Prove that

for all and

Answer:

Let

for all and

For , L.H.S

R.H.S

Therefore LHS = RHS

Hence is true for

Let is true for

… i)

Now we have to show that is true for

is true for

Hence by the principle of mathematical induction

Question 35: Prove that

for all natural numbers,

Answer:

Let for all natural numbers,

For , L.H.S R.H.S Therefore LHS = RHS

Hence is true for

Let is true for

for all natural numbers, … … i)

Now we have to show that is true for

is true for

Hence by the principle of mathematical induction

for all natural numbers,

Question 36: Prove that for all

Answer:

Let for all

For ,

Hence is true for

Let is true for

for all … … … … … i)

Now we have to show that is true for

Since

Since

is true for

Hence by the principle of mathematical induction

for all

Question 37: for all

Answer:

Let for all

For ,

Hence is true for

Let is true for

for all … … … … … i)

Now we have to show that is true for

is true for

Hence by the principle of mathematical induction

for all

Question 38: Prove that is divisible by for all

Answer:

Let is divisible by for all

For which is divisible by

Hence is true for

Let is true for

is divisible by for all

… … … … i)

Now we have to show that is true for

where

is true for

Hence by the principle of mathematical induction

is divisible by for all

Question 39: Prove that for all

Answer:

Let for all

For latex P(1) $ is true for

Let is true for

for all … … … i)

Now we have to show that is true for

is true for

Hence by the principle of mathematical induction

for all

Question 40: Prove that

Answer:

Let

For

RHS

Therefore LHS = RHS

Hence is true for

Let is true for

… i)

Now we have to show that is true for

is true for

Hence by the principle of mathematical induction

Question 41: Prove that , for all natural numbers

Answer:

Let , for all natural numbers

For , L.H.S R.H.S

Hence is true for

Let is true for

, for all natural numbers … … … … … i)

Now we have to show that is true for

is true for

Hence by the principle of mathematical induction

, for all natural numbers

Question 42: Give

and

for , where

Prove that

Answer:

Let

For LHS

RHS

Therefore LHS = RHS

Hence is true for

Let is true for

is true for … … … … … i)

Now we have to show that is true for

LHS

RHS

Therefore LHS = RHS

is true for

Hence by the principle of mathematical induction

Question 43: Let be the statement: . If is true, show that is true. Do you conclude that is true for all ?

Answer:

Let for all

For , LHS RHS

Therefore LHS RHS

Hence is NOT true for

Therefore is NOT true for all

Question 44: Show that by the principle of mathematical induction that the sum of the terms of the series is given by

Answer:

Let

of the terms of the series is given by

For ,

L.H.S

R.H.S

Therefore LHS = RHS

Hence is true for

For ,

L.H.S

R.H.S

Therefore LHS = RHS

Hence is true for

Let is true for

of the terms of the series is given by

… … … … … i)

Now we have to show that is true for

If is even, is odd

If is odd, is even

is true for

Hence by the principle of mathematical induction

of the terms of the series is given by

Question 45: Prove that the number of subsets of a set containing distinct elements is for all

Answer:

Let P(n) : The number of subsets of a set containing distinct elements is for all

For LHS number of subsets of a set containing only element and the set itself RHS .

Hence is true for

Let is true for

The number of subsets of a set containing distinct elements is for all

Now we have to show that is true for

Let

Now

Using i) we can say that has subset and as subset.

has subsets.

is true for

Hence by the principle of mathematical induction The number of subsets of a set containing distinct elements is for all

Question 46: A sequence is defined by letting and for all natural numbers . Show that for all .

Answer:

Given a sequence is defined by letting and for all natural numbers

Let for all

For , L.H.S

Hence is true for

Let is true for

for all … … i)

Now we have to show that is true for

is true for

Hence by the principle of mathematical induction for all

Question 47: A sequence is defined by letting and for all natural numbers . Show that for all .

Answer:

Given a sequence is defined by letting and for all natural numbers .

Let for all .

For , L.H.S

Hence is true for

Let is true for

for all … … i)

Now we have to show that is true for

is true for

Hence by the principle of mathematical induction for all

Question 48: A sequence is defined by letting and for all natural numbers . Show that for all using mathematical induction.

Answer:

Given A sequence is defined by letting and for all natural numbers .

Let for all

For , L.H.S

Hence is true for

Let is true for

for all … … i)

Now we have to show that is true for

is true for

Hence by the principle of mathematical induction

for all

Question 49: Using the principle of mathematical induction prove that

for all natural numbers .

Answer:

Let for all natural numbers

For , L.H.S R.H.S Therefore LHS < RHS

Hence is true for

Let is true for

for all natural numbers … … i)

Now we have to show that is true for

LHS

RHS

Since

Therefore LHS RHS

is true for

Hence by the principle of mathematical induction

for all natural numbers

Question 50: The distributive law from algebra states that for all real numbers and we have . Use this law and mathematical induction to prove that, for all natural numbers, , if are any real numbers, then

Answer:

For all real numbers , and

To prove that, for all natural numbers, , if are any real numbers, then

Let for all natural numbers , and

For , LHS , RHS LHS RHS

Hence is true for

Let is true for therefore for all natural numbers , and

For

Therefore is true for

Hence by the principle of mathematical induction

for all natural numbers , and