Question 26:

Answer:

… … … … … i)

We know that .

Hence by the principle of mathematical induction

Question 27:

Answer:

… … … … … i)

where

Hence by the principle of mathematical induction

Question 28:

Answer:

R.H.S

… … i)

Hence by the principle of mathematical induction

Question 29:

Answer:

… … … … … i)

where

Hence by the principle of mathematical induction

Question 30:

Answer:

R.H.S

… … … … … i)

Hence by the principle of mathematical induction

Question 31:

Answer:

R.H.S

… … … … … i)

Hence by the principle of mathematical induction

Question 32: Prove that

Answer:

… … … … … i)

Hence by the principle of mathematical induction

Answer:

,

L.H.S

1 is an integer.

Which is an integer.

Hence by the principle of mathematical induction

Question 34: Prove that

for all

Answer:

for all

R.H.S

Therefore LHS = RHS

… i)

Hence by the principle of mathematical induction

Question 35: Prove that

for all natural numbers,

Answer:

for all natural numbers,

R.H.S

for all natural numbers, … … i)

Hence by the principle of mathematical induction

for all natural numbers,

Answer:

,

… … … … … i)

Since

Since

Hence by the principle of mathematical induction

Answer:

,

… … … … … i)

Hence by the principle of mathematical induction

Question 38: Prove that

Answer:

… … … … i)

where

Hence by the principle of mathematical induction

Answer:

Therefore LHS = RHS

… … … i)

Hence by the principle of mathematical induction

Question 40: Prove that

Answer:

Therefore LHS = RHS

… i)

Hence by the principle of mathematical induction

Answer:

, for all natural numbers

R.H.S

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

Hence by the principle of mathematical induction

, for all natural numbers

Question 42: Give

Answer:

LHS

Therefore LHS = RHS

… … … … … i)

Therefore LHS = RHS

Hence by the principle of mathematical induction

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

Answer:

, LHS RHS

Therefore LHS RHS

is NOT true

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:

of the terms of the series is given by

,

Therefore LHS = RHS

,

Therefore LHS = RHS

of the terms of the series is given by

… … … … … i)

If is even, is odd

If is odd, is even

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

Answer:

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

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

The number of subsets of a set containing distinct elements is

Now

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

has subsets.

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

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

Answer:

Given a sequence is defined by letting for all natural numbers

… … i)

Hence by the principle of mathematical induction

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

Answer:

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

.

… … i)

Hence by the principle of mathematical induction

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

Answer:

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

… … i)

Hence by the principle of mathematical induction

Question 49: Using the principle of mathematical induction prove that

Answer:

for all natural numbers

R.H.S Therefore LHS < RHS

for all natural numbers … … i)

LHS

RHS

Since

Therefore LHS RHS

Hence by the principle of mathematical induction

for all natural numbers

Question 50: The distributive law from algebra states that for all real numbers 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

for all natural numbers , and

, LHS , RHS LHS RHS

therefore for all natural numbers , and

Therefore

Hence by the principle of mathematical induction

for all natural numbers , and