{"id":4589,"date":"2016-05-17T09:49:00","date_gmt":"2016-05-17T04:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-11-maths-exercise-4-1\/"},"modified":"2016-05-17T10:44:14","modified_gmt":"2016-05-17T05:14:14","slug":"ncert-solutions-class-11-maths-exercise-4-1","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-11-maths-exercise-4-1\/","title":{"rendered":"NCERT Solutions class-11 Maths Exercise 4.1"},"content":{"rendered":"

Exercise 4.1<\/strong><\/p>\n

Prove the following by using the principle of mathematical induction for all N:<\/strong><\/p>\n

1. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

1 = 1<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

<\/p>\n

\u2026\u2026\u2026.(i)<\/p>\n

For [Using eq. (i)]\n

<\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

2. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

1 = 1<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

\u2026\u2026\u2026.(i)<\/p>\n

For <\/p>\n

<\/p>\n

<\/p>\n

= <\/p>\n

= <\/p>\n

<\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

3. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

1 = 1<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

\u2026\u2026\u2026.(i)<\/p>\n

For <\/p>\n

[Using (i)]\n

<\/p>\n

= <\/p>\n

= <\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

4. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

6 = 6<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

\u2026\u2026\u2026(i)<\/p>\n

For <\/p>\n

<\/p>\n

[Using eq. (i)]\n

<\/p>\n

= <\/p>\n

<\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

5. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

3 = 3<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

<\/p>\n

For <\/p>\n

<\/p>\n

<\/p>\n

= <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

= <\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

6. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

2 = 2<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

\u2026\u2026\u2026(i)<\/p>\n

For <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

= <\/p>\n

= <\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

7. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

3 = 3<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

<\/p>\n

For <\/p>\n

<\/p>\n

= <\/p>\n

<\/p>\n

= <\/p>\n

<\/p>\n

= <\/p>\n

= <\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

8. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

2 = 2<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

<\/p>\n

For <\/p>\n

<\/p>\n

<\/p>\n

= <\/p>\n

= = <\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

9. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

<\/p>\n

For <\/p>\n

<\/p>\n

<\/p>\n

= = <\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

10. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

<\/p>\n

For <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

= <\/p>\n

= <\/p>\n

<\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

11. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

\u2026\u2026\u2026.(i)<\/p>\n

For <\/p>\n

R.H.S. = <\/p>\n

And L.H.S. = [Using eq. (i)]\n

= <\/p>\n

= <\/p>\n

= <\/p>\n

= <\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

12. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

\u2026\u2026\u2026.(i)<\/p>\n

For <\/p>\n

R.H.S. = <\/p>\n

L.H.S. = [Using eq. (i)]\n

L.H.S. = <\/p>\n

= <\/p>\n

= <\/p>\n

= <\/p>\n

= <\/p>\n

= <\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

13. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

\u2026\u2026\u2026.(i)<\/p>\n

For <\/p>\n

R.H.S. = <\/p>\n

L.H.S. = [Using eq. (i)]\n

L.H.S. = <\/p>\n

= <\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

14. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

<\/p>\n

For R.H.S. = <\/p>\n

L.H.S. = [Using eq. (i)]\n

L.H.S. = = <\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

15. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

\u2026\u2026\u2026.(i)<\/p>\n

For R.H.S. = <\/p>\n

L.H.S. = [Using eq. (i)]\n

= <\/p>\n

= <\/p>\n

= <\/p>\n

= <\/p>\n

= <\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

16. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

\u2026\u2026\u2026.(i)<\/p>\n

For R.H.S. = <\/p>\n

L.H.S. = <\/p>\n

L.H.S. = <\/p>\n

= <\/p>\n

= <\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

17. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

\u2026\u2026\u2026.(i)<\/p>\n

For R.H.S. = <\/p>\n

L.H.S. = <\/p>\n

L.H.S. = <\/p>\n

= <\/p>\n

= <\/p>\n

= <\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

18. <\/strong><\/p>\n

Ans. <\/strong>Let <\/p>\n

For <\/p>\n

<\/p>\n

is true.<\/p>\n

Now, let be true for <\/p>\n

\u2026\u2026\u2026.(i)<\/p>\n

For , <\/p>\n

Now, adding on both sides of eq. (i), we have<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

8 < 9<\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

19. <\/strong> is a multiple of 3.<\/strong><\/p>\n

Ans. <\/strong>Let is a multiple of 3.<\/p>\n

For 1 (1 + 1) (1 + 5) is a multiple of 3 = 12 is a multiple of 3<\/p>\n

P (1) is true.<\/p>\n

Let be true for , is a multiple of 3.<\/p>\n

<\/p>\n

<\/p>\n

\u2026.(i)<\/p>\n

For , is a multiple of 3<\/p>\n

Now, <\/p>\n

= <\/p>\n

= [Using (i)]\n

= <\/p>\n

= <\/p>\n

= is a multiple of 3<\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

20. <\/strong> is divisible by 11.<\/strong><\/p>\n

Ans. <\/strong>Let is divisible by 11.<\/p>\n

For is divisible by 11<\/p>\n

= 11 is divisible by 11<\/p>\n

P (1) is true.<\/p>\n

Let be true for , is divisible by 11 = <\/p>\n

\u2026\u2026\u2026.(i)<\/p>\n

For is divisible by 11<\/p>\n

is divisible by 11<\/p>\n

Now, <\/p>\n

= <\/p>\n

= <\/p>\n

is divisible by 11<\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

21. <\/strong> is divisible by <\/strong><\/p>\n

Ans. <\/strong>Let is divisible by <\/p>\n

For is divisible by = is divisible by <\/p>\n

P (1) is true.<\/p>\n

Let be true for , is divisible by = <\/p>\n

\u2026\u2026\u2026.(i)<\/p>\n

For is divisible by <\/p>\n

Now, <\/p>\n

= <\/p>\n

= <\/p>\n

= [From eq. (i)]\n

= <\/p>\n

is divisible by <\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

22. <\/strong> is divisible by 8.<\/strong><\/p>\n

Ans. <\/strong>Let is divisible by 8.<\/p>\n

For is divisible by 8 = 64 is divisible by 8<\/p>\n

P (1) is true.<\/p>\n

Let be true for , is divisible by 8 = <\/p>\n

\u2026\u2026\u2026.(i)<\/p>\n

For is divisible by 8<\/p>\n

is divisible by 8<\/p>\n

Now, <\/p>\n

= [From eq. (i)]\n

= <\/p>\n

= = <\/p>\n

is divisible by 8<\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

23. <\/strong> is a multiple of 27.<\/strong><\/p>\n

Ans. <\/strong>Let is a multiple of 27.<\/p>\n

For is a multiple of 27 = 27 is a multiple of 27<\/p>\n

P (1) is true.<\/p>\n

Let be true for , is a multiple of 27 = \u2026..(i)<\/p>\n

For is a multiple of 27<\/p>\n

Now, <\/p>\n

= <\/p>\n

= <\/p>\n

= [From eq. (i)]\n

= <\/p>\n

is a multiple of 27<\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n

\n

24. <\/strong><\/p>\n

Ans.<\/strong> Let <\/p>\n

For <\/p>\n

<\/p>\n

9 < 16<\/p>\n

P (1) is true.<\/p>\n

Let be true for <\/p>\n

\u2026\u2026\u2026.(i)<\/p>\n

For <\/p>\n

<\/p>\n

= <\/p>\n

Now, adding 2 on both sides in eq. (i),<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

Also <\/p>\n

<\/p>\n

is true.<\/p>\n

Therefore, is true.<\/p>\n

is true.<\/p>\n

Hence by Principle of Mathematical Induction, is true for all N.<\/p>\n","protected":false},"excerpt":{"rendered":"

Exercise 4.1 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 \u2026\u2026\u2026.(i) For [Using eq. (i)] is true. Therefore, is true. is true. Hence by Principle of Mathematical Induction, is true for all N. 2. Ans. Let … Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[47,281],"tags":[],"class_list":["post-4589","post","type-post","status-publish","format-standard","hentry","category-cbse-class-11","category-ncert-solutions"],"yoast_head":"\nNCERT Solutions class-11 Maths Exercise 4.1 | myCBSEguide<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-11-maths-exercise-4-1\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"NCERT Solutions class-11 Maths Exercise 4.1 | myCBSEguide\" \/>\n<meta property=\"og:description\" content=\"Exercise 4.1 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 \u2026\u2026\u2026.(i) For [Using eq. (i)] is true. Therefore, is true. is true. Hence by Principle of Mathematical Induction, is true for all N. 2. Ans. Let ... Read more\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-11-maths-exercise-4-1\/\" \/>\n<meta property=\"og:site_name\" content=\"myCBSEguide\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/mycbseguide\/\" \/>\n<meta property=\"article:published_time\" content=\"2016-05-17T04:19:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2016-05-17T05:14:14+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch04\/Ex4.1\/image001.png\" \/>\n<meta name=\"author\" content=\"myCBSEguide\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@mycbseguide\" \/>\n<meta name=\"twitter:site\" content=\"@mycbseguide\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"myCBSEguide\" \/>\n\t<meta name=\"twitter:label2\" 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