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Exercise 4.1 - Chapter 4 Principle Of Mathematical Induction class 11 ncert solutions Maths - SaraNextGen [2024]


Question 1:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4096/chapter%204_html_7b648934.gif

 

Answer:

Let the given statement be P(n), i.e.,

P(n): 1 + 3 + 32 + …+ 3n–1 =https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4096/chapter%204_html_2176954f.gif

For n = 1, we have

P(1): 1 =https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4096/chapter%204_html_m2e255c98.gif, which is true.

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4096/chapter%204_html_m14a0dd51.gif

We shall now prove that P(k + 1) is true.

Consider

1 + 3 + 32 + … + 3k–1 + 3(k+1) – 1

= (1 + 3 + 32 +… + 3k–1) + 3k

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4096/chapter%204_html_md426c7.gif 

 

 

 

 

 

 

 

 

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 2:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4097/chapter%204_html_2869c332.gif

Answer:

Let the given statement be P(n), i.e.,

P(n): https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4097/chapter%204_html_2869c332.gif

For n = 1, we have

P(1): 13 = 1 =https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4097/chapter%204_html_m17983561.gif, which is true.

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4097/chapter%204_html_6ee980f9.gif

We shall now prove that P(k + 1) is true.

Consider

13 + 23 + 33 + … + k3 + (k + 1)3

= (13 + 23 + 33 + …. + k3) + (k + 1)3 https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4097/chapter%204_html_7af62ec3.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 3:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4098/chapter%204_html_3a2eaa8c.gif

Answer:

Let the given statement be P(n), i.e.,

P(n): https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4098/chapter%204_html_54ff5917.gif

For n = 1, we have

P(1): 1 =https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4098/chapter%204_html_m204519e9.gif which is true.

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4098/chapter%204_html_m198c70.gif

We shall now prove that P(k + 1) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4098/chapter%204_html_m2076dfe9.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4098/chapter%204_html_6208f1e4.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

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) =https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4099/chapter%204_html_3888f331.gif

Answer:

Let the given statement be P(n), i.e.,

P(n): 1.2.3 + 2.3.4 + … + n(n + 1) (n + 2) =https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4099/chapter%204_html_3888f331.gif

For n = 1, we have

P(1): 1.2.3 = 6 =https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4099/chapter%204_html_m7319837d.gif, which is true.

Let P(k) be true for some positive integer k, i.e.,

1.2.3 + 2.3.4 + … + k(k + 1) (k + 2) https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4099/chapter%204_html_m4efbcbee.gif

We shall now prove that P(k + 1) is true.

Consider

1.2.3 + 2.3.4 + … + k(k + 1) (k + 2) + (k + 1) (k + 2) (k + 3)

= {1.2.3 + 2.3.4 + … + k(k + 1) (k + 2)} + (k + 1) (k + 2) (k + 3)

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4099/chapter%204_html_18874a56.gif 

 

 

 

 

 

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 5:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4100/chapter%204_html_m2902b327.gif

Answer:

Let the given statement be P(n), i.e.,

P(n) : https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4100/chapter%204_html_m16749061.gif

For n = 1, we have

P(1): 1.3 = 3https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4100/chapter%204_html_m1604a2b7.gif, which is true.

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4100/chapter%204_html_m4dd7bf1c.gif

We shall now prove that P(k + 1) is true.

Consider

1.3 + 2.32 + 3.33 + … + k3k+ (k + 1) 3k+1

= (1.3 + 2.32 + 3.33 + …+ k.3k) + (k + 1) 3k+1

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4100/chapter%204_html_m6de89cf4.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 6:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4101/chapter%204_html_5b1b9907.gif

Answer:

Let the given statement be P(n), i.e.,

P(n): https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4101/chapter%204_html_5b1b9907.gif

For n = 1, we have

P(1): https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4101/chapter%204_html_48b077a8.gif, which is true.

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4101/chapter%204_html_15a080fa.gif

We shall now prove that P(k + 1) is true.

Consider

1.2 + 2.3 + 3.4 + … + k.(+ 1) + (k + 1).(k + 2)

= [1.2 + 2.3 + 3.4 + … + k.(k + 1)] + (k + 1).(k + 2)

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4101/chapter%204_html_1fd1b699.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 7:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4102/chapter%204_html_32de5642.gif

Answer:

Let the given statement be P(n), i.e.,

P(n): https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4102/chapter%204_html_32de5642.gif

For n = 1, we have

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4102/chapter%204_html_m5646804c.gif, which is true.

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4102/chapter%204_html_m75fc59c8.gif

We shall now prove that P(k + 1) is true.

Consider

(1.3 + 3.5 + 5.7 + … + (2k – 1) (2k + 1) + {2(k + 1) – 1}{2(k + 1) + 1}

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4102/chapter%204_html_12450083.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4102/chapter%204_html_m61479ca9.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

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

Answer:

Let the given statement be P(n), i.e.,

P(n): 1.2 + 2.22 + 3.22 + … + n.2n = (n – 1) 2n+1 + 2

For n = 1, we have

P(1): 1.2 = 2 = (1 – 1) 21+1 + 2 = 0 + 2 = 2, which is true.

Let P(k) be true for some positive integer k, i.e.,

1.2 + 2.22 + 3.22 + … + k.2k = (k – 1) 2k + 1 + 2 … (i)

We shall now prove that P(k + 1) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4103/chapter%204_html_2db7408e.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 9:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4104/chapter%204_html_28c5704b.gif

Answer:

Let the given statement be P(n), i.e.,

P(n): https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4104/chapter%204_html_28c5704b.gif

For n = 1, we have

P(1): https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4104/chapter%204_html_2532c7e5.gif, which is true.

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4104/chapter%204_html_m295aef22.gif

We shall now prove that P(k + 1) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4104/chapter%204_html_37e82c58.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 10:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4105/chapter%204_html_m76583aa0.gif

Answer:

Let the given statement be P(n), i.e.,

P(n): https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4105/chapter%204_html_m76583aa0.gif

For n = 1, we have

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4105/chapter%204_html_m6d6af1e3.gif, which is true.

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4105/chapter%204_html_5db0a1f3.gif

We shall now prove that P(k + 1) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4105/chapter%204_html_m7e13b464.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4105/chapter%204_html_m25fc40c0.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 11:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4106/chapter%204_html_6b495eca.gif

Answer:

Let the given statement be P(n), i.e.,

P(n): https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4106/chapter%204_html_6b495eca.gif

For n = 1, we have

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4106/chapter%204_html_fd15574.gif, which is true.

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4106/chapter%204_html_7a7788ce.gif

We shall now prove that P(k + 1) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4106/chapter%204_html_m1b9dcde.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4106/chapter%204_html_632df9ea.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 12:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4107/chapter%204_html_m3b5c4261.gif

Answer:

Let the given statement be P(n), i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4107/chapter%204_html_d8eb225.gif

For n = 1, we have

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4107/chapter%204_html_m603bb1e.gif, which is true.

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4107/chapter%204_html_2da2a144.gif

We shall now prove that P(k + 1) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4107/chapter%204_html_10d796e6.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 13:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4108/chapter%204_html_m75fcb1d4.gif

Answer:

Let the given statement be P(n), i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4108/chapter%204_html_mbc21083.gif

For n = 1, we have

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4108/chapter%204_html_m6c7a591d.gif

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4108/chapter%204_html_270299de.gif

We shall now prove that P(k + 1) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4108/chapter%204_html_1782e21d.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 14:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4109/chapter%204_html_m7d82e8e9.gif

Answer:

Let the given statement be P(n), i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4109/chapter%204_html_3cf5d8f8.gif

For n = 1, we have

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4109/chapter%204_html_1a542736.gif, which is true.

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4109/chapter%204_html_27f848ed.gif

We shall now prove that P(k + 1) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4109/chapter%204_html_m567e4be0.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 15:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4110/chapter%204_html_m57f0433a.gif

Answer:

Let the given statement be P(n), i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4110/chapter%204_html_2abc64d0.gif

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4110/chapter%204_html_4eb7f853.gif

We shall now prove that P(k + 1) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4110/chapter%204_html_m49f277b1.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4110/chapter%204_html_m5ba4bc5f.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 16:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4111/chapter%204_html_m6202e6e9.gif

Answer:

Let the given statement be P(n), i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4111/chapter%204_html_m56ee6e3d.gif

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4111/chapter%204_html_1eafbf10.gif

We shall now prove that P(k + 1) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4111/chapter%204_html_m82b2937.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 17:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4112/chapter%204_html_3eff732d.gif

Answer:

Let the given statement be P(n), i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4112/chapter%204_html_m255f7cc3.gif

For n = 1, we have

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4112/chapter%204_html_e09466a.gif, which is true.

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4112/chapter%204_html_m9bb79d3.gif

We shall now prove that P(k + 1) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4112/chapter%204_html_m71c66296.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 18:

Prove the following by using the principle of mathematical induction for all n ∈ Nhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4113/chapter%204_html_m415ed4a5.gif

Answer:

Let the given statement be P(n), i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4113/chapter%204_html_544be049.gif

It can be noted that P(n) is true for n = 1 since https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4113/chapter%204_html_7dd1284.gif.

Let P(k) be true for some positive integer k, i.e.,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4113/chapter%204_html_12d1d117.gif

We shall now prove that P(k + 1) is true whenever P(k) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4113/chapter%204_html_m3797162e.gif

Hence,https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4113/chapter%204_html_m1f0476a5.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 19:

Prove the following by using the principle of mathematical induction for all n ∈ Nn (n + 1) (n + 5) is a multiple of 3.

Answer:

Let the given statement be P(n), i.e.,

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

It can be noted that P(n) is true for n = 1 since 1 (1 + 1) (1 + 5) = 12, which is a multiple of 3.

Let P(k) be true for some positive integer k, i.e.,

k (k + 1) (k + 5) is a multiple of 3.

k (k + 1) (k + 5) = 3m, where m ∈ N … (1)

We shall now prove that P(k + 1) is true whenever P(k) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4114/chapter%204_html_m65ae3a69.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 20:

Prove the following by using the principle of mathematical induction for all n ∈ N: 102n – 1 + 1 is divisible by 11.

Answer:

Let the given statement be P(n), i.e.,

P(n): 102n – 1 + 1 is divisible by 11.

It can be observed that P(n) is true for n = 1 since P(1) = 102.1 – 1 + 1 = 11, which is divisible by 11.

Let P(k) be true for some positive integer k, i.e.,

102k – 1 + 1 is divisible by 11.

∴102k – 1 + 1 = 11m, where m ∈ … (1)

We shall now prove that P(k + 1) is true whenever P(k) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4115/chapter%204_html_2b2e983a.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 21:

Prove the following by using the principle of mathematical induction for all n ∈ Nx2n – y2n is divisible by x y.

Answer:

Let the given statement be P(n), i.e.,

P(n): x2n – y2n is divisible by x y.

It can be observed that P(n) is true for n = 1.

This is so because x2 × 1 – y2 × 1 = x2 – y2 = (y) (x – y) is divisible by (x + y).

Let P(k) be true for some positive integer k, i.e.,

x2k – y2k is divisible by x y.

x2k – y2k = m (y), where m ∈ N … (1)

We shall now prove that P(k + 1) is true whenever P(k) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4116/chapter%204_html_m3f92156c.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 22:

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

Answer:

Let the given statement be P(n), i.e.,

P(n): 32n + 2 – 8n – 9 is divisible by 8.

It can be observed that P(n) is true for n = 1 since 32 × 1 + 2 – 8 × 1 – 9 = 64, which is divisible by 8.

Let P(k) be true for some positive integer k, i.e.,

32k + 2 – 8k – 9 is divisible by 8.

∴32k + 2 – 8k – 9 = 8m; where m ∈ N … (1)

We shall now prove that P(k + 1) is true whenever P(k) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4117/chapter%204_html_5a03f2fb.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 23:

Prove the following by using the principle of mathematical induction for all n ∈ N: 41n – 14n is a multiple of 27.

Answer:

Let the given statement be P(n), i.e.,

P(n):41n – 14nis a multiple of 27.

It can be observed that P(n) is true for n = 1 since https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4118/chapter%204_html_4d4c97ba.gifwhich is a multiple of 27.

Let P(k) be true for some positive integer k, i.e.,

41k – 14kis a multiple of 27

∴41k – 14k = 27m, where m ∈ N … (1)

We shall now prove that P(k + 1) is true whenever P(k) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/4118/chapter%204_html_70776182.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Question 24:

Prove the following by using the principle of mathematical induction for allhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/5541/Chapter%204_html_m162593da.gif

(2+7) < (n + 3)2

Answer:

Let the given statement be P(n), i.e.,

P(n): (2+7) < (n + 3)2

It can be observed that P(n) is true for n = 1 since 2.1 + 7 = 9 < (1 + 3)2 = 16, which is true.

Let P(k) be true for some positive integer k, i.e.,

(2k + 7) < (k + 3)2 … (1)

We shall now prove that P(k + 1) is true whenever P(k) is true.

Consider

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/164/5541/Chapter%204_html_m58c954a4.gif

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

Also Read : Exercise-5.1-Chapter-5-Complex-Numbers-&-Quadratic-Equations-class-11-ncert-solutions-Maths

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