Prove the following by the principle of mathematical induction:

Question 1: i.e., the sum of the first natural numbers is .

Answer:

Let

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

L.H.S R.H.S Hence is true for

Let is true for

… … … … … i)

Now we have to show that is true for

Now substituting the from equation i)

is true for

Hence by the principle of mathematical induction

is true for all

Question 2:

Answer:

Let

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

L.H.S R.H.S Hence is true for

Let is true for

… … … … … i)

Now we have to show that is true for

Now substituting the from equation i)

is true for

Hence by the principle of mathematical induction

is true for all

Question 3:

Answer:

Let

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

L.H.S R.H.S Hence is true for

Let is true for

… … … … … i)

Now we have to show that is true for

Now substituting the from equation i)

is true for

Hence by the principle of mathematical induction

is true for all

Question 4:

Answer:

Let

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

L.H.S R.H.S Hence is true for

Let is true for

… … … … … i)

Now we have to show that is true for

Now substituting the from equation i)

is true for

Hence by the principle of mathematical induction

is true for all

Question 5: i.e., the sum of firs odd numbers is

Answer:

Let

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

L.H.S R.H.S Hence is true for

Let is true for

… … … … … i)

Now we have to show that is true for

Now substituting the from equation i)

is true for

Hence by the principle of mathematical induction

is true for all

Question 6:

Answer:

Let

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

L.H.S R.H.S 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

is true for all

Question 7:

Answer:

Let

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

L.H.S R.H.S 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

is true for all

Question 8:

Answer:

Let

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

L.H.S R.H.S Hence is true for

Let is true for

… … … … … i)

Now we have to show that is true for

Question 9:

Answer:

Let

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

L.H.S R.H.S 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

is true for all

Question 10:

Answer:

Let

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

L.H.S R.H.S 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

is true for all

Question 11:

Answer:

Let

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

L.H.S R.H.S 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

is true for all

Question 12:

Answer:

Let

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

L.H.S R.H.S 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

is true for all

Question 13:

Answer:

Let

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

L.H.S R.H.S 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

is true for all

Question 14:

Answer:

Let

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

L.H.S R.H.S 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

is true for all

Question 15:

Answer:

Let

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

L.H.S R.H.S 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

is true for all

Question 16:

Answer:

Let

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

L.H.S R.H.S 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

is true for all

Question 17:

Answer:

Let

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

L.H.S R.H.S 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

is true for all

Question 18:

Answer:

Let

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

L.H.S R.H.S 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

is true for all

Question 19: is divisible by for all

Answer:

Let is divisible by for all

For , L.H.S which is divisible by 24

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 20: 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 21: 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 22: 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 23: for all

Answer:

Let for all

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

Hence is true for

Let is true for

for all

Now we have to show that is true for

is true for

Hence by the principle of mathematical induction

for all

Question 24: is a multiple of for all

Answer:

Let is a multiple of 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 a multiple of for all

Question 25: 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