NCERT Solutions Class 11 Mathematics Chapter 4 Download In Pdf

Chapter 4 Principle of Mathematical Induction Download in pdf

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

**Question **1. 1 + 3 + 32 + ... + 3n â€“ 1 =
(3 1)
2
n âˆ’
.

**Question 2. **13 + 23 + 33 + â€¦ +n3 =
2 ( 1)
2
n n +
.

**Question 3.** 1 1 1 1 2
(1 2) (1 2 3) (1 2 3 ) ( 1)
... n
...n n
+ + + + =
+ + + + + + + .

**Question 4.** 1.2.3 + 2.3.4 +â€¦+ n(n+1) (n+2) =
( 1)( 2)( 3)
4
n n + n + n +
.

**Question 5.** 1.3 + 2.32 + 3.33 +â€¦+ n.3n =
(2 1)3 1 3
4
n âˆ’ n+ +
.

**Question 6. **1.2 + 2.3 + 3.4 +â€¦+ n.(n+1) =
( 1)( 2)
3
n n + n +.

**Question 7.** 1.3 + 3.5 + 57 +â€¦+ (2nâ€“1) (2n+1) =
(4 2 6 1)
3
n n + n âˆ’
.

**Question 8. **1.2 + 2.22 + 3.22 + ...+n.2n = (nâ€“1) 2n + 1 + 2.

**Question 9.** 1 1 1 ... 1 1 1
2 4 8 2n 2n + + + + = âˆ’ .

**Question 10.** 1 1 1 ... 1
2.5 5.8 8.11 (3 1)(3 2) (6 4)
n
n n n
+ + + + =
âˆ’ + + .

**Question 11.** 1 1 1 ... 1 ( 3)
1.2.3 2.3.4 3.4.5 ( 1)( 2) 4( 1)( 2)

**Question 12.** a + ar + ar2 +â€¦+ arn-1 =
( 1)
1
a rn
r
âˆ’
âˆ’
.

**Question 13.** 2
2
1 3 1 5 1 7 ... 1 (2 1) ( 1)
1 4 9
n n

**Question 14. **12 + 32 + 52 + â€¦+ (2nâ€“1)2 =
(2 1)(2 1)
3
n n âˆ’ n +
.

**Question 15.** 1 1 1 ... 1
1.4 4.7 7.10 (3 2)(3 1) (3 1)
n
n n n
+ + + + =
âˆ’ + + .

**Question 16.** 1 1 1 ... 1
3.5 5.7 7.9 (2 1)(2 3) 3(2 3)
n
n n n
+ + + + =
+ + + .

**Question 17.** 1 + 2 + 3 +â€¦+ n <
1
8
(2n + 1)2.

**Question 18. **n (n + 1) (n + 5) is a multiple of 3.

**Question 19. **102n â€“ 1 + 1 is divisible by 11.

**Question 20. **x2n â€“ y2n is divisible by x + y.

**Question 21. **32n+2 â€“ 8n â€“ 9 is divisible by 8.

**Question 22.** 41n â€“ 14n is a multiple of 27.

**Question 23."> (2n + 7) < (n + 3)2.
**

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- Chapter 1 Sets
- chapter 2 Relations And Functions
- Chapter 3 Trigonometric Functions
- Chapter 4 Principle of Mathematical Induction
- Chapter 5 Complex Numbers and Quadratic Equations
- Chapter 6 Linear Inequalities
- Chapter 7 Permutations And Combinations
- Chapter 8 Binomial Theorem
- Chapter 9 Sequences And Series
- Chapter 10 Straight Lines
- Chapter 11 Conic Sections
- Chapter 12 Introduction To Three Dimensional Geometry
- Chapter 13 Limits and Derivatives
- Chapter 14 Mathematical Reasoning
- Chapter 15 Statistics
- Chapter 16 Probability

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