Principle of Mathematical Induction - NCERT Solutions

 CBSE Class–11 Mathematics

NCERT Solutions
Chapter - 4 Principle of Mathematical Induction
Exercise 4.1

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

1. 

Ans. Let 

For 

 

1 = 1

 is true.

Now, let  be true for 

  ……….(i)

For 

  [Using eq. (i)]

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n)  is true for all  N.

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2. 

Ans. Let 

For 

 

1 = 1

 is true.

Now, let  be true for 

……….(i)

For 

= 

= 

 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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3. 

Ans. Let 

For 

1 = 1

 is true.

Now, let  be true for 

 ……….(i)

For 

 [Using (i)]

= 

= 

 is true.

Therefore,  is true..

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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4. 

Ans. Let 

For 

6 = 6

 is true.

Now, let  be true for 

  ………(i)

For 

 [Using eq. (i)]

= 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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5. 

Ans. Let 

For 

3 = 3

 is true.

Now, let  be true for 

For 

 

= 

= 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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6.  

Ans. Let 

For 

2 = 2

 is true.

Now, let  be true for 

  ………(i)

For 

= 

= 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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7. 

Ans. Let 

For 

3 = 3

 is true.

Now, let  be true for 

For 

= 

= 

 

= 

= 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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8. 

Ans. Let 

For 

2 = 2

 is true.

Now, let  be true for 

For 

= 

=  = 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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9. 

Ans. Let 

For 

P(1) is true.

Now, let  P(n) be true for 

For 

=  = 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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10. 

Ans. Let 

For 

 is true.

Now, let  be true for 

 

For 

 

 P(k+1)=k6k+4+1(3k+2)(3k+5)$P(k+1)=\frac{k}{6k+4}+\frac{1}{(3k+2)(3k+5)}$

= 

= 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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11. 

Ans. Let 

For 

 is true.

Now, let  be true for 

……….(i)

For 

R.H.S. = 

And L.H.S. =  [Using eq. (i)]

= 

= 

= 

= 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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12. 

Ans. Let 

For 

 is true.

Now, let  be true for 

 ……….(i)

For 

R.H.S. = 

L.H.S. =a(rk−1)r−1+ark$\frac{a(r^k-1)}{r-1}+a{r}^k$   [Using eq. (i)]

L.H.S. = 

= 

= 

= 

= 

= 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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13. 

Ans. Let 

For 

 is true.

Now, let  be true for 

  ……….(i)

For 

R.H.S. = 

L.H.S. = [Using eq. (i)]

L.H.S. = 

= 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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14. 

Ans. Let 

For 

 is true.

Now, let  be true for 

 

For 

R.H.S. = 

L.H.S. = [Using eq. (i)]

L.H.S. =  = 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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15. 

Ans. Let 

For 

 

 is true.

Now, let  be true for 

  ……….(i)

For 

R.H.S. = 

L.H.S. = [Using eq. (i)]

=(2k+1)(k(2k−1)3+(2k+1))$(2k+1)\left(\frac{k(2k-1)}{3}+(2k+1)\right)$

= (2k+1)(2k2−k+6k+3)3)$(2k+1)\left(\frac{2k^2-k+6k+3)}{3}\right)$

= 

= 

= 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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16. 

Ans. Let 

For 

 is true.

Now, let  be true for 

  ……….(i)

For 

R.H.S. = 

L.H.S. = 

L.H.S. = 

= 

= 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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17. 

Ans. Let 

For 

 is true.

Now, let  be true for 

 ……….(i)

For 

R.H.S. = 

L.H.S. = 

 L.H.S. = 

= 

= 

= 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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18. 

Ans. Let 

For 

 

 is true.

Now, let  be true for 

 ……….(i)

For ,

  

Now, adding  on both sides of eq. (i), we have

 8 < 9

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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19.  is a multiple of 3.

Ans. Let  is a multiple of 3.

For 

  1 (1 + 1) (1 + 5) is a multiple of 3 = 12 is a multiple of 3

P (1) is true.

Let  be true for ,

 is a multiple of 3.

 ….(i)

For ,

  is a multiple of 3

Now, 

= 

=  [Using (i)]

= 

= 

=  is a multiple of 3

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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20.  is divisible by 11.

Ans. Let  is divisible by 11.

For  is divisible by 11

= 11 is divisible by 11

P (1) is true.

Let  be true for ,

  is divisible by 11

 

 ……….(i)

For 

  is divisible by 11

  is divisible by 11

Now, 

= 

= 

 is divisible by 11

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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21.  is divisible by 

Ans. Let  is divisible by 

For 

 is divisible by 

  is divisible by 

P (1) is true.

Let  be true for ,

  is divisible by 

 

 ……….(i)

For 

 is divisible by 

Now,

 x2k+2−y2k+2=x2k+2−x2ky2+x2ky2−y2k+2$x^{2k+2}-y^{2k+2}=x^{2k+2}-x^{2k}{y}^2+x^{2k}{y}^2-y^{2k+2}$

= 

= 

=  [From eq. (i)]

= 

 is divisible by 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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22.  is divisible by 8.

Ans. Let  is divisible by 8.

For 

 is divisible by 8

 64 is divisible by 8

P (1) is true.

Let  be true for ,

 is divisible by 8

 

 ……….(i)

For 

 is divisible by 8

  is divisible by 8

Now, 

=  [From eq. (i)]

= 

= 

 

 is divisible by 8

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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23.  is a multiple of 27.

Ans. Let  is a multiple of 27.

For  is a multiple of 27

 27 is a multiple of 27

P (1) is true.

Let  be true for ,

  is a multiple of 27

 …..(i)

For 

 is a multiple of 27

Now, 

= 

= 

=  [From eq. (i)]

= 

 is a multiple of 27

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.

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24. 

Ans. Let 

For 

9 < 16

P (1) is true.

Let  be true for 

 ……….(i)

For 

= 

Now, adding 2 on both sides in eq. (i),

Also 

 is true.

Therefore,  is true.

Hence by Principle of Mathematical Induction, P(n) is true for all  N.