#### Chapters

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Values of Trigonometric function at sum or difference of angles

Chapter 8: Transformation formulae

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Chapter 10: Sine and cosine formulae and their applications

Chapter 11: Trigonometric equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 14: Quadratic Equations

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progression

Chapter 20: Geometric Progression

Chapter 21: Some special series

Chapter 22: Brief review of cartesian system of rectangular co-ordinates

Chapter 23: The straight lines

Chapter 24: The circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to three dimensional coordinate geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical reasoning

Chapter 32: Statistics

Chapter 33: Probability

#### RD Sharma Mathematics Class 11

## Chapter 21: Some special series

#### Chapter 21: Some special series Exercise 21.10 solutions [Page 10]

1^{3 }+ 3^{3 }+ 5^{3} + 7^{3} + ...

2^{2} + 4^{2} + 6^{2} + 8^{2} + ...

1.2.5 + 2.3.6 + 3.4.7 + ...

1.2.4 + 2.3.7 +3.4.10 + ...

1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + ...

1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + ...

3 × 1^{2} + 5 ×2^{2} + 7 × 3^{2} + ...

Find the sum of the series whose nth term is:

2n^{2} − 3n + 5

Find the sum of the series whose nth term is:

2n^{3} + 3n^{2} − 1

Find the sum of the series whose nth term is:

n^{3} − 3^{n}

Find the sum of the series whose nth term is:

n (n + 1) (n + 4)

Find the sum of the series whose nth term is:

(2n − 1)^{2}

Find the 20^{th} term and the sum of 20 terms of the series 2 × 4 + 4 × 6 + 6 × 8 + ...

#### Chapter 21: Some special series Exercise 21.20 solutions [Page 18]

3 + 5 + 9 + 15 + 23 + ...

2 + 5 + 10 + 17 + 26 + ...

1 + 3 + 7 + 13 + 21 + ...

3 + 7 + 14 + 24 + 37 + ...

1 + 3 + 6 + 10 + 15 + ...

1 + 4 + 13 + 40 + 121 + ...

4 + 6 + 9 + 13 + 18 + ...

2 + 4 + 7 + 11 + 16 + ...

\[\frac{1}{1 . 4} + \frac{1}{4 . 7} + \frac{1}{7 . 10} + . . .\]

\[\frac{1}{1 . 6} + \frac{1}{6 . 11} + \frac{1}{11 . 14} + \frac{1}{14 . 19} + . . . + \frac{1}{(5n - 4) (5n + 1)}\]

#### Chapter 21: Some special series solutions [Pages 18 - 19]

Write the sum of the series 2 + 4 + 6 + 8 + ... + 2n.

Write the sum of the series 1^{2} − 2^{2} + 3^{2} − 4^{2} + 5^{2} − 6^{2} + ... + (2n − 1)^{2} − (2n)^{2}.

Write the sum to *n* terms of a series whose *r*^{th} term is *r* + 2^{r}.

If \[\sum^n_{r = 1} r = 55, \text{ find } \sum^n_{r = 1} r^3\] .

If the sum of first *n* even natural numbers is equal to *k* times the sum of first *n* odd natural numbers, then write the value of *k*.

Write the sum of 20 terms of the series \[1 + \frac{1}{2}(1 + 2) + \frac{1}{3}(1 + 2 + 3) + . . . .\]

Write the 50th term of the series 2 + 3 + 6 + 11 + 18 + ...

Let *S*_{n} denote the sum of the cubes of first n natural numbers and s_{n} denote the sum of first n natural numbers. Then, write the value of \[\sum^n_{r = 1} \frac{S_r}{s_r}\] .

#### Chapter 21: Some special series solutions [Pages 19 - 20]

The sum to n terms of the series \[\frac{1}{\sqrt{1} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{7}} + . . . . + . . . .\] is

\[\sqrt{2n + 1}\]

\[\frac{1}{2}\sqrt{2n + 1}\]

\[\sqrt{2n + 1} - 1\]

\[\frac{1}{2}\left\{ \sqrt{2n + 1} - 1 \right\}\]

The sum of the series

\[\frac{1}{\log_2 4} + \frac{1}{\log_4 4} + \frac{1}{\log_8 4} + . . . . + \frac{1}{\log_2^n 4}\] is

\[\frac{n (n + 1)}{2}\]

\[\frac{n (n + 1) (2n + 1)}{12}\]

\[\frac{n (n + 1)}{4}\]

none of these

The value of \[\sum^n_{r = 1} \left\{ (2r - 1) a + \frac{1}{b^r} \right\}\] is equal to

\[a n^2 + \frac{b^{n - 1} - 1}{b^{n - 1} (b - 1)}\]

\[a n^2 + \frac{b^n - 1}{b^n (b - 1)}\]

\[a n^3 + \frac{b^{n - 1} - 1}{b^n (b - 1)}\]

none of these

If ∑ n = 210, then ∑ n^{2} =

2870

2160

2970

none of these

If *S*_{n} = \[\sum^n_{r = 1} \frac{1 + 2 + 2^2 + . . . \text { Sum to r terms }}{2^r}\], then *S*_{n} is equal to

2

^{n}− n − 1\[1 - \frac{1}{2^n}\]

\[n - 1 + \frac{1}{2^n}\]

2

^{n}− 1

If \[1 + \frac{1 + 2}{2} + \frac{1 + 2 + 3}{3} + . . . .\] to n terms is S, then S is equal to

\[\frac{n (n + 3)}{4}\]

\[\frac{n (n + 2)}{4}\]

\[\frac{n (n + 1) (n + 2)}{6}\]

n

^{2}

Sum of n terms of the series \[\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} +\] ....... is

\[\frac{n (n + 1)}{2}\]

2n (n + 1)

\[\frac{n (n + 1)}{\sqrt{2}}\]

1

The sum of 10 terms of the series \[\sqrt{2} + \sqrt{6} + \sqrt{18} +\] .... is

\[121 (\sqrt{6} + \sqrt{2})\]

\[243 (\sqrt{3} + 1)\]

\[\frac{121}{\sqrt{3} - 1}\]

\[242 (\sqrt{3} - 1)\]

The sum of the series 1^{2} + 3^{2} + 5^{2} + ... to n terms is

\[\frac{n (n + 1) (2n + 1)}{2}\]

\[\frac{n (2n - 1) (2n + 1)}{3}\]

\[\frac{(n - 1 )^2 (2n + 1)}{6}\]

\[\frac{(2n + 1 )^3}{3}\]

The sum of the series \[\frac{2}{3} + \frac{8}{9} + \frac{26}{27} + \frac{80}{81} +\] to n terms is

\[n - \frac{1}{2}( 3^{- n} - 1)\]

\[n - \frac{1}{2}(1 - 3^{- n} )\]

\[n + \frac{1}{2}( 3^n - 1)\]

\[n - \frac{1}{2}( 3^n - 1)\]

## Chapter 21: Some special series

#### RD Sharma Mathematics Class 11

#### Textbook solutions for Class 11

## RD Sharma solutions for Class 11 Mathematics chapter 21 - Some special series

RD Sharma solutions for Class 11 Maths chapter 21 (Some special series) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics chapter 21 Some special series are Relationship Between A.M. and G.M., Geometric Progression (G. P.), Arithmetic Progression (A.P.), Concept of Series, Concept of Sequences, Introduction of Sequence and Series, Sum to N Terms of Special Series.

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