**Prove the following by using the principle of mathematical induction for all n ∈ N:**

**Question****1.**1+3+3^{2}+…..+3^{n-1}=(3^{n}-1)/2**Solution :**

Letp(n):1+3+3^{2}+…..+3^{n-1} = (3^{n}-1)/2

for n = 1

L.H.S = 3^{1-1} = 1

Hence by Principle of Mathematical Induction,N is true for all n ∈ N.

**Question****2.**

Solution :

**Question****3.**Prove the following by using the principle of mathematical induction for all n∈N .

**Solution :**

**Question****4. Prove the following by using the principle of mathematical induction for all n∈ N:**

**1.2.3 +2.3.4 +…+ n(n + 1)(n + 2) = n(n+1) (n+2) (n+3)/4**

**Solution :**

Therefore, P(k+1) holds whenever P(k)

holds.

Hence, the given equality is true for all natural numbers i.e., N

by the principle of mathematical induction.

**Question****5. ****Prove the following by using the principle of mathematical induction for all n∈ N:**

**Solution :**

**Question****6. Prove the following by using the principle of mathematical induction for all n∈ N:**

Hence, the given equality is true for all natural numbers i.e., N by the principle of mathematical induction.

.

**Question****7. **Prove the following by using the principle of mathematical induction for all n∈ N:

**Solution :**

**Question 8.** Prove the following by using the principle of mathematical induction for all n∈ N:

1.2+2.2^{2}+3.2^{2}+….+n.2^{n}=(n-1)2^{n+1}+2

**Solution :**

Let p(n) : 1.2+2.2^{2}+3.2^{2}+….+n.2^{n}=(n-1)2^{n+1}+2

For n = 1

**L.H.S **=**1.2** = **2**

R.H.S.=(1-1)2^{1+1}+2=0+2=2

Now, let p(n) be true for n = 1

Let us assume that P(k) is true for some positive integer k, i.e.,

1.2+2.2^{2}+3.2^{2}+…+k.2^{k}=(k-1)2^{k+1}+2…(i)

Now, we have to prove that P(k+1) is also true.

Consider

{1.2+2.2^{2}+3.2^{2}+…+k.2^{k}}+(k+1).^{2k+1}

=(k-1)^{2k+1}+2+(k+1)^{2k+1}

=2^{k+1}{(k-1)+(k+1)}+2

=2^{k+1}.2k+2

=k.2^{(k+1)+1 }+2

={(k+1)-1}2^{(k+1)+1}+2

=Therefore, P(k+1) holds whenever P(k)

holds.

Hence, the given equality is true for all natural numbers i.e., N

by the principle of mathematical induction.

Hence by Principle of Mathematical Induction,is true for all n ∈ N.

**Question 9. **Prove the following by using the principle of mathematical induction for all n∈ N:

**Question 10. **Prove the following by using the principle of mathematical induction for all n∈ N:

**Solution :**

Let

For n = 1

**Question****11. **Prove the following by using the principle of mathematical induction for all n∈ N:

**Solution :**

**Question****12. **Prove the following by using the principle of mathematical induction for all n∈ N:

**Solution :**

**Question****13. **Prove the following by using the principle of mathematical induction for all n∈ N:

**Solution :**

**Question****14.** Prove the following by using the principle of mathematical induction for all n∈ N:

Solution :

Let

**Question 15.** Prove the following by using the principle of mathematical induction for all n∈ N:

**Solution :**

**Question****16.**

**Solution :**

** for n = 1**

Therefore, P(k+1) holds whenever P(k)holds.

Hence, the given equality is true for all natural numbers i.e., N

by the principle of mathematical induction.

**Question**** 17. Prove the following by using the principle of mathematical induction for alln∈N:**

**Solution :**

Therefore, P(k+1) holds whenever P(k)holds.

Hence, the given equality is true for all natural numbers i.e., N

by the principle of mathematical induction.

**Question****18. Prove the following by using the principle of mathematical induction for alln∈N:**

**Solution :**

**Question****19. Prove the following by using the principle of mathematical induction for alln∈N:n (n + 1) (n + 5) is a multiple of 3.**

**Solution :**Let us denote the given statement by P(n)

i.e.

P(n):n(n+1)(n+5) which is a multiple of 3

For n=1

1(1+1)(1+5)=12,

which is a multiple of 3.

Therefore, P(n)

is true for n=1.

Let us assume that P(k)

is true for some natural number k,

k(k+1)(k+5)

is a multiple of 3.

∴k(k+1)(k+5)=3m

, where m∈N

…(i)

Now, we have to prove that P(k+1)

is also true whenever P(k)

is true.

Consider

(k+1){(k+1)+1}{(k+1)+5}

=(k+1)(k+2){(k+1)+5}

=(k+1)(k+2)(k+5)+(k+1)(k+2)

={k(k+1)(k+5)+2(k+1)(k+5)}+(k+1)(k+2)

=3m+(k+1){2(k+5)+(k+2)}

=3m+(k+1){2k+10+k+2}

=3m+(k+1){3k+12}

=3m+3(k+1){k+4}

=3{m+(k+1)(k+4)}=3 × q

, where q={m+(k+1)(k+4)}

is some natural number.

Hence, (k+1){(k+1)+1}{(k+1)+5}

is a multiple of 3

.

Therefore, P(k+1)

holds whenever P(k)

holds.

Hence, the given equality is true for all natural numbers i.e., N

by the principle of mathematical induction.

**Question****20. Prove the following by using the principle of mathematical induction for alln∈N: 10 ^{2n-1 }+ 1 is divisible by 11.**

**Ans.**

**Let P(n) : 10 ^{2n-1 }+ 1 is divisible by 11.**

For n = 1 is divisible by 11

P(1)=**10 ^{2n-1 }+ 1**=11 and P(1) is divisible by 11

Therefore, P(n)

is true for n=1

Let us assume that P(k)

is true for some natural number k

i.e.,

i.e., 10

^{2n-1}+1

is divisible by 11

.

∴10

^{2k-1}+1 = 11m

, where m∈N

…(i)

Now, we have to prove that P(k+1)

is also true whenever P(k)

is true.

Consider

10

^{2(k+1) -1}+ 1

=10

^{2k+2-1}+1

=10

^{2k+1}+1

=10

^{2(102k-1+1-1)}+1

=10

^{2}(10

^{2k-1}+1 -1)-10

^{2}+1

=10

^{2}.11m-100+1 Using(i)

=100 × 11m-99

=11(100m-9)

=11r

, where r=(100m-9)

is some natural number

Therefore, 10

^{2(k+1)-1}+1

is divisible by 11

Therefore, P(k+1)

holds whenever P(k)

holds.

Hence, the given equality is true for all natural numbers i.e., N

by the principle of mathematical induction.

**Question 21. Prove the following by using the principle of mathematical induction for alln∈N: x ^{2n }–y^{2}n is divisible by x+y.**

**Solution :**

Hence, the given equality is true for all natural numbers i.e., N by the principle of mathematical induction.true.

**Question****22. Prove the following by using the principle of mathematical induction for all n∈N: 3 ^{2n+2}-8n-9is divisible by 8.**

**Solution :**

Let** 3 ^{2n+2}-8n-9** is divisible by 8.

For** p(n)**: 3is divisible by 8 = 64 is divisible by 8

=8r, where r=(9m+8k+8) is a natural number

Therefore,

32(k+1)+2-8(k+1)-9

is divisible by 8

Therefore, P(k+1) holds whenever P(k)

holds.

Hence, the given equality is true for all natural numbers i.e., N

by the principle of mathematical induction.

**Question 23. Prove the following by using the principle of mathematical induction for all n∈N: 41 ^{n }– 14^{n }is a multiple of 27.**

**Solution :**

Let** 41 ^{n }– 14^{n }**is a multiple of 27.

for n = 1,

Hence, the given equality is true for all natural numbers i.e., N by the principle of mathematical induction.

**Question 24.(2n+7) < (n + 3) ^{2}**

**Solution :**