CLASS 11 MATHS CHAPTER-4 PRINCIPLE OF MATHEMATICAL INDUCTION

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

Question1.1+3+3²+…..+3^n-1=(3ⁿ-1)/2


Solution :
Letp(n):1+3+3²+…..+3^n-1 = (3ⁿ-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.

Question2.

Solution :

Question3.Prove the following by using the principle of mathematical induction for all n∈N .

Solution :

Question4. 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.

Question5. Prove the following by using the principle of mathematical induction for all n∈ N:

Solution :

Question6. 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.

.

Question7. 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+2.2²+3.2²+….+n.2ⁿ=(n-1)2ⁿ⁺¹+2

Solution :
Let p(n) : 1.2+2.2²+3.2²+….+n.2ⁿ=(n-1)2ⁿ⁺¹+2

For n = 1

L.H.S =1.2 = 2

R.H.S.=(1-1)2¹⁺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+2.2²+3.2²+…+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+2.2²+3.2²+…+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

Question11. Prove the following by using the principle of mathematical induction for all n∈ N:

Solution :

Question12. Prove the following by using the principle of mathematical induction for all n∈ N:

Solution :

Question13. Prove the following by using the principle of mathematical induction for all n∈ N:

Solution :

Question14. 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 :

Question16.

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.

Question18. Prove the following by using the principle of mathematical induction for alln∈N:

Solution :

Question19. 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.

Question20. 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

 

=10²(10^2k-1+1 -1)-10²+1

=10².11m-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: x²ⁿ –y²n 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.

Question22. Prove the following by using the principle of mathematical induction for all n∈N: 3²ⁿ⁺²-8n-9
is divisible by 8.

Solution :
Let 3²ⁿ⁺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: 41ⁿ – 14ⁿ is a multiple of 27.

Solution :
Let 41ⁿ – 14ⁿ 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 :