RevisionDojo

Introduction

Sequences and series form a crucial part of the JEE Main Mathematics syllabus. They find applications in various areas of mathematics and are foundational for understanding more advanced topics. This study note will break down the concepts of sequences and series into manageable sections, providing detailed explanations and examples to ensure a thorough understanding.

Sequences

A sequence is an ordered list of numbers, where each number is called a term. Sequences can be finite or infinite.

Types of Sequences

Arithmetic Sequence (AP)

An arithmetic sequence is a sequence in which the difference between consecutive terms is constant. This difference is called the common difference ($d$).

General Form: $a, a+d, a+2d, a+3d, \ldots$

nth Term:

$$ a_n = a + (n-1)d $$

Example

Example: Find the 10th term of the sequence 2, 5, 8, 11, ...

Here, $a = 2$ and $d = 3$.

$$ a_{10} = 2 + (10-1) \cdot 3 = 2 + 27 = 29 $$

Geometric Sequence (GP)

A geometric sequence is a sequence where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio ($r$).

General Form: $a, ar, ar^2, ar^3, \ldots$

nth Term:

$$ a_n = ar^{n-1} $$

Example

Example: Find the 6th term of the sequence 3, 6, 12, 24, ...

Here, $a = 3$ and $r = 2$.

$$ a_6 = 3 \cdot 2^{5} = 3 \cdot 32 = 96 $$

Harmonic Sequence (HP)

A harmonic sequence is a sequence of numbers whose reciprocals form an arithmetic sequence.

If the sequence is: $a, b, c, \ldots$

Then its reciprocals form an AP: $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \ldots$

Tip

To solve problems involving harmonic sequences, convert them to their corresponding arithmetic sequence of reciprocals.

Special Sequences

Fibonacci Sequence

A Fibonacci sequence is a sequence where each term is the sum of the two preceding ones, usually starting with 0 and 1.

General Form: $0, 1, 1, 2, 3, 5, 8, \ldots$

nth Term:

$$ F_n = F_{n-1} + F_{n-2} $$

Example

Example: Find the 7th term of the Fibonacci sequence.

$$ F_7 = F_6 + F_5 = 8 + 5 = 13 $$

Series

A series is the sum of the terms of a sequence.

Arithmetic Series (AP)

The sum of the first $n$ terms of an arithmetic sequence is called an arithmetic series.

Sum of first $n$ terms:

$$ S_n = \frac{n}{2} [2a + (n-1)d] $$

Example

Example: Find the sum of the first 10 terms of the sequence 2, 5, 8, 11, ...

Here, $a = 2$, $d = 3$, and $n = 10$.

$$ S_{10} = \frac{10}{2} [2 \cdot 2 + (10-1) \cdot 3] = 5 [4 + 27] = 5 \cdot 31 = 155 $$

Geometric Series (GP)

The sum of the first $n$ terms of a geometric sequence is called a geometric series.

Sum of first $n$ terms (when $r \neq 1$):

$$ S_n = a \frac{r^n - 1}{r - 1} $$

Sum of an infinite geometric series (when $|r|

< 1$):

$$ S = \frac{a}{1 - r} $$

Example

Example: Find the sum of the first 5 terms of the sequence 3, 6, 12, 24, ...

Here, $a = 3$, $r = 2$, and $n = 5$.

$$ S_5 = 3 \frac{2^5 - 1}{2 - 1} = 3 \frac{32 - 1}{1} = 3 \cdot 31 = 93 $$

Important Concepts

Arithmetic Mean (AM)

The arithmetic mean of two numbers $a$ and $b$ is given by:

$$ AM = \frac{a + b}{2} $$

Geometric Mean (GM)

The geometric mean of two numbers $a$ and $b$ is given by:

$$ GM = \sqrt{ab} $$

Harmonic Mean (HM)

The harmonic mean of two numbers $a$ and $b$ is given by:

$$ HM = \frac{2ab}{a + b} $$

Note

The relationship between AM, GM, and HM for any two positive numbers $a$ and $b$ is:

$$ AM \geq GM \geq HM $$

Sum to Infinity

For an infinite geometric series with $|r|

< 1$, the sum to infinity is:

$$ S = \frac{a}{1 - r} $$

Common Mistake

A common mistake is to apply the sum to infinity formula to series where $|r| \geq 1$. Ensure the common ratio $r$ satisfies $|r|

< 1$ before using the formula.

Practice Problems

Find the 15th term of the arithmetic sequence 7, 10, 13, ...

Determine the sum of the first 8 terms of the geometric sequence 5, 15, 45, ...

Calculate the harmonic mean of 4 and 9.

If the sum of the first 20 terms of an arithmetic sequence is 210, find the common difference if the first term is 2.

Find the sum to infinity of the geometric series 4, 2, 1, ...

Conclusion

Understanding sequences and series is essential for solving various mathematical problems in the JEE Main examination. By mastering the concepts and practicing regularly, you can develop a strong foundation in this topic. Remember to pay attention to the common pitfalls and use the tips provided to enhance your problem-solving skills.

Unlock the rest of this chapter with a Free account

Nice try, unfortunately this paywall isn't as easy to bypass as you think. Want to help devleop the site? Join the team at https://revisiondojo.com/join-us. exercitation voluptate cillum ullamco excepteur sint officia do tempor Lorem irure minim Lorem elit id voluptate reprehenderit voluptate laboris in nostrud qui non Lorem nostrud laborum culpa sit occaecat reprehenderit

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Duis aute irure dolor in reprehenderit

Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit aut fugit, sed quia consequuntur magni dolores eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci velit.

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Introduction

Sequences

A sequence is an ordered list of numbers, where each number is called a term. Sequences can be finite or infinite.

Types of Sequences

Arithmetic Sequence (AP)

An arithmetic sequence is a sequence in which the difference between consecutive terms is constant. This difference is called the common difference ($d$).

General Form: $a, a+d, a+2d, a+3d, \ldots$

nth Term:

$$ a_n = a + (n-1)d $$

Example

Example: Find the 10th term of the sequence 2, 5, 8, 11, ...

Here, $a = 2$ and $d = 3$.

$$ a_{10} = 2 + (10-1) \cdot 3 = 2 + 27 = 29 $$

Geometric Sequence (GP)

A geometric sequence is a sequence where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio ($r$).

General Form: $a, ar, ar^2, ar^3, \ldots$

nth Term:

$$ a_n = ar^{n-1} $$

Example

Example: Find the 6th term of the sequence 3, 6, 12, 24, ...

Here, $a = 3$ and $r = 2$.

$$ a_6 = 3 \cdot 2^{5} = 3 \cdot 32 = 96 $$

Harmonic Sequence (HP)

A harmonic sequence is a sequence of numbers whose reciprocals form an arithmetic sequence.

If the sequence is: $a, b, c, \ldots$

Then its reciprocals form an AP: $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \ldots$

Tip

To solve problems involving harmonic sequences, convert them to their corresponding arithmetic sequence of reciprocals.

Special Sequences

Fibonacci Sequence

A Fibonacci sequence is a sequence where each term is the sum of the two preceding ones, usually starting with 0 and 1.

General Form: $0, 1, 1, 2, 3, 5, 8, \ldots$

nth Term:

$$ F_n = F_{n-1} + F_{n-2} $$

Example

Example: Find the 7th term of the Fibonacci sequence.

$$ F_7 = F_6 + F_5 = 8 + 5 = 13 $$

Series

A series is the sum of the terms of a sequence.

Arithmetic Series (AP)

The sum of the first $n$ terms of an arithmetic sequence is called an arithmetic series.

Sum of first $n$ terms:

$$ S_n = \frac{n}{2} [2a + (n-1)d] $$

Example

Example: Find the sum of the first 10 terms of the sequence 2, 5, 8, 11, ...

Here, $a = 2$, $d = 3$, and $n = 10$.

$$ S_{10} = \frac{10}{2} [2 \cdot 2 + (10-1) \cdot 3] = 5 [4 + 27] = 5 \cdot 31 = 155 $$

Geometric Series (GP)

The sum of the first $n$ terms of a geometric sequence is called a geometric series.

Sum of first $n$ terms (when $r \neq 1$):

$$ S_n = a \frac{r^n - 1}{r - 1} $$

Sum of an infinite geometric series (when $|r|

< 1$):

$$ S = \frac{a}{1 - r} $$

Example

Example: Find the sum of the first 5 terms of the sequence 3, 6, 12, 24, ...

Here, $a = 3$, $r = 2$, and $n = 5$.

$$ S_5 = 3 \frac{2^5 - 1}{2 - 1} = 3 \frac{32 - 1}{1} = 3 \cdot 31 = 93 $$

Important Concepts

Arithmetic Mean (AM)

The arithmetic mean of two numbers $a$ and $b$ is given by:

$$ AM = \frac{a + b}{2} $$

Geometric Mean (GM)

The geometric mean of two numbers $a$ and $b$ is given by:

$$ GM = \sqrt{ab} $$

Harmonic Mean (HM)

The harmonic mean of two numbers $a$ and $b$ is given by:

$$ HM = \frac{2ab}{a + b} $$

Note

The relationship between AM, GM, and HM for any two positive numbers $a$ and $b$ is:

$$ AM \geq GM \geq HM $$

Sum to Infinity

For an infinite geometric series with $|r|

< 1$, the sum to infinity is:

$$ S = \frac{a}{1 - r} $$

Common Mistake

A common mistake is to apply the sum to infinity formula to series where $|r| \geq 1$. Ensure the common ratio $r$ satisfies $|r|

< 1$ before using the formula.

Practice Problems

Find the 15th term of the arithmetic sequence 7, 10, 13, ...

Determine the sum of the first 8 terms of the geometric sequence 5, 15, 45, ...

Calculate the harmonic mean of 4 and 9.

If the sum of the first 20 terms of an arithmetic sequence is 210, find the common difference if the first term is 2.

Find the sum to infinity of the geometric series 4, 2, 1, ...

Conclusion

Unlock the rest of this chapter with a Free account

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Algebra11 subtopics

Calculus8 subtopics

Coordinate Geometry5 subtopics

Trigonometry4 subtopics

Differential Calculus1 subtopic

Sequences And Series Notes

Introduction

Sequences

Types of Sequences

Arithmetic Sequence (AP)

Geometric Sequence (GP)

Harmonic Sequence (HP)

Special Sequences

Fibonacci Sequence

Series

Arithmetic Series (AP)

Geometric Series (GP)

Important Concepts

Arithmetic Mean (AM)

Geometric Mean (GM)

Harmonic Mean (HM)

Sum to Infinity

Practice Problems

Conclusion

Unlock the rest of this chapter with a Free account

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Algebra11 subtopics

Calculus8 subtopics

Coordinate Geometry5 subtopics

Trigonometry4 subtopics

Differential Calculus1 subtopic

Introduction

Sequences

Types of Sequences

Arithmetic Sequence (AP)

Geometric Sequence (GP)

Harmonic Sequence (HP)

Special Sequences

Fibonacci Sequence

Series

Arithmetic Series (AP)

Geometric Series (GP)

Important Concepts

Arithmetic Mean (AM)

Geometric Mean (GM)

Harmonic Mean (HM)

Sum to Infinity

Practice Problems

Conclusion

Unlock the rest of this chapter with a Free account

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident