Question 1: Let be the statement divides . What is ?

Answer:

is the statement divides .

Clearly, is obtained by replacing by in .

Question 2: If is the statement , prove that whenever is true, is also true.

Answer:

The statement is . Let be true. Then

We wish to prove that the statement is true i.e.

Now,

is true

[ adding on both sides]

is true [ for every natural number ]

Thus, whenever is true, is also true.

Question 3: Let be the statement . If is true, prove that is true.

Answer:

We are given that is true i.e. and we wish to prove that is true i.e. .

Now,

is true

[Multiplying both sides by ]

[ for every . for every ]

is true

Question 4: lf is the statement is an integral multiple of , and if is true, prove that is true.

Answer:

Let be true. Then, is an integral multiple of .

We wish to prove that is true i.e. is an integral multiple of .

Now,

is true

. is an integral multiple of

, for some

Now,

, where

is an integral multiple of

is true

Question 5: Prove by the principle of mathematical induction that for all is even natural number.

Answer:

Let be the statement is even

We have, is even

, which is even

is true

Let be true. Then,

is ture is even for some

Now, we shall show thal is true. For this we have to show that is an even natural number.

,where

is an even natural number

is true

Thus, is true is true

Hence,by the principle of mathematical induction, is true for all i.e. is even for all .

Question 6: Prove by the principle of mathematical induction that is divisible by for all

Answer:

Let be the statement is divisible by

i.e. is divisible by

We have, is divisible by

which is divisibleby

) is true

Let be true. Then,

is divisible by

, for some

Now, we shall show that is true. For this we have to show that is divisibleby

which is divisible by

is true

Thus, is true is true

Hence, by the principle of mathematical induction, the given statement is true for all

Question 7: Prove by induction that the sum is divisible by for all

Answer:

Let be the statement given by is divisible by

Wehave, is divisible by

is true

Let be true. Then,

is divisible by

for

We now wish to show that is true. For this we have to show that is divisible by

Now,

where

is true

Thus, is true is true

Hence, by the principle of mathematical induction the statement is true for all

Question 8: Using principle of mathematical induction, prove that it divisible by for all

Answer:

Let be the statement given by is divisible by

which is divisible by

So, is true.

Let be true. Then,

is divisible by

We shall now show that is true i.e. is divisible by

Now

Clearly, it is divisible by

is true

Thus, is true is true.

Hence , by the principle of mathematical induction is true for all

i.e. it divisible by for all

Question 9: Using the principle of mathematical induction, prove that is divisible by for all

Answer:

Let be the statement given by is divisible by

is divisible by

Clearly, which is divisibte by

So, is true

Let be true. Then,

is divisible by

for

We shall now show that is true. For this we have to show that is divisible by

Now, is divisible by

is true

is true is true

Hence , by the principle of mathematical induction, is true for all is divisible by for all

Question 10: Prove that: is divisible by , for all

Answer:

Let be the statement given by is divisible by , for all

We have, which is divisible by

is true

Let be true. Then, is divisible by , for all

for some

We shall now show that is true. For this we have to show that is divisible by

[ Since is a multiple of for all for all ]

which is divisible by

Therefore is true

Thus, is true is true

Hence, by the principle of mathematical induction, is true for all

Question 11: Prove by induction that for all natural numbers . Using this, prove by induction that for all

Answer:

Let be the statement given by

Therefore is true

Let be true. Then, for all natural numbers

We shall now show that is true whenever is true. For this we have to show that

Now, is true

is true

Hence, by the principle of mathematical induction is true for all

Let be the statement given by for all

is true

Let be true. Then, for all natural numbers

We shall now show that is true whenever is true. For this we have to show that

Now, is true

is true

Hence by the principle of mathematical induction is true for all i.e. for all