Sequence and Series

A sequence refers to a structured arrangement of elements in a specific order that follows a specific pattern or rule. These elements can be numbers, letters, or any other objects. In mathematics, sequences are fundamental concepts that are extensively used in various areas, such as calculus, algebra, and number theory.

A series represents the cumulative total obtained by adding together the terms of a sequence. Series are essential in mathematics and have wide applications in calculus, physics, engineering, and many other fields.

Understanding sequences and series is crucial in mathematics as they form the basis for calculus, which deals with concepts like limits, derivatives, and integrals. They also play a vital role in solving mathematical problems, modeling real-world phenomena, and developing mathematical theories and techniques.

1.0Sequence and Series Definition

A sequence refers to a structured arrangement of elements in a specific order that follows a specific pattern or rule. Each term in a sequence is obtained by applying a consistent rule to the preceding term. For example, the Sequence of whole numbers (0, 1, 2, 3, 4, ...) is formed by adding 1 to each preceding term.

A series represents the cumulative total obtained by adding together the terms of a sequence. It is formed by adding all the terms of a sequence together. For instance, the sum of the first n natural numbers is an example of a series, denoted as 1 + 2 + 3 + ... + n.

In summary, a sequence is a list of terms following a pattern, while a series is the sum of these terms. Both sequences and series are fundamental concepts in mathematics and are widely used in various applications, including calculus, number theory, and finance.

2.0Types of Sequence and Series 

There are several types of sequences and series in mathematics, each with its own characteristics and properties. Here are some common types of Sequence

1. Arithmetic Sequence:

In an arithmetic sequence, each term is generated by adding a constant value (known as the common difference) to the preceding term.

Example: 2, 5, 8, 11, 14, ...

2. Geometric Sequence:

G. P is a sequence of non-zero numbers. Each of the succeeding terms is equal to the preceding term multiplied by a constant. Thus, in a GP, the ratio of the successive terms is constant (called the common ratio). 

Example: 2, 6, 18, 54, …

3. Harmonic Sequence:

In a harmonic sequence, the reciprocals of the terms form an arithmetic sequence.

Example: 1, $\frac{1}{2},\frac{1}{4},\frac{3}{4},...$

4. Fibonacci Sequence:

The Fibonacci sequence is a distinctive series where each number is the sum of the two preceding numbers. It commences with 0 and 1, and the following terms are formed by adding the two previous numbers.

Example: 0, 1, 1, 2, 3, 5, 8, 13, and so forth.

Common Types of Series

1. Arithmetic Series:

An arithmetic series refers to the summation of terms within an arithmetic sequence.

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

2. Geometric Series:

A geometric series represents the total of terms within a geometric sequence.

Example: Sum of the first 5 terms of the sequence 2, 6, 18, 54, ...

3. Infinite Series:

An infinite series is a series that extends endlessly without a definite endpoint, either with a fixed pattern or following a specific rule.

Example: Sum of the infinite series $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...$

4. Convergent and Divergent Series:

A convergent series is characterized by its sum gradually approaching a finite value as more terms are added.

A divergent series is one whose sum does not approach a finite value as the number of terms increases.

3.0Difference between Sequence and Series

Sequence

1. A sequence refers to a structured arrangement of elements in a specific order or numbers arranged according to a rule or pattern.
2. It is represented as a₁, a₂, a₃​, …, a_n, where a₁, a₂, a₃, … are the individual terms or elements of the sequence.
3. A sequence can be finite (having a specific last term) or infinite (continuing indefinitely).

Examples of sequences include arithmetic sequences, geometric sequences, Fibonacci sequences, etc.

The focus in a sequence is on the arrangement and order of the elements based on a specific rule or pattern.

Series

1. A series represents the cumulative total obtained by adding together the terms of a sequence.
2. It can be represented as S_n = a₁ + a₂ + a₃ + … + a_n, where S_n represents the sum of the initial n terms of the Sequence.
3. A series can also be finite (summing a specific number of terms) or infinite (summing an infinite number of terms).

Examples of series include arithmetic series, geometric series, infinite series, etc.

4. The focus in a series is on finding the total sum or value obtained by adding the terms of a sequence.

4.0Sequence and Series Formula

Here are some common formulas related to sequences and series:

• Arithmetic Sequence:

a_n = a₁ + (n−1)⋅d - n^th term of an arithmetic sequence

${\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}\cdot \left({\mathrm{a}}_1+{\mathrm{a}}_{\mathrm{n}}\right)$- sum of the first n terms of an arithmetic sequence

${\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}\cdot \left(2{\mathrm{a}}_1+(\mathrm{n}-1)\cdot \mathrm{d}\right)$- sum of an arithmetic series

• Geometric Sequence:

a_n​ = a₁​⋅rⁿ⁻¹ - n^th term of a geometric sequence

${\mathrm{S}}_{\mathrm{n}}={\mathrm{a}}_1\cdot \frac{\left({\mathrm{r}}_{\mathrm{n}}-1\right)}{\mathrm{r}-1}$- sum of the first n terms of a geometric sequence

$\mathrm{S}=\frac{{\mathrm{a}}_1}{1-\mathrm{r}}$- sum of an infinite geometric series (when ∣r∣<1)

• Harmonic Series:

${\mathrm{S}}_{\mathrm{n}}=1+\frac{1}{2}+\frac{1}{3}+\dots +\frac{1}{\mathrm{n}}$ ​- sum of the first n terms of the harmonic series (n is finite)

• Fibonacci Sequence:

F_n = F_n−1 + F_n−2 - n^th term of the Fibonacci sequence (starting with F₁ = 1, F₂ = 1 or F₀ = 0, F₁ = 1)

5.0Sequence and Series Jee Mains Question

1. Let S_n denote the sum of the initial n terms of an arithmetic progression. If S₁₀ = 390 and the ratio of the tenth and the fifth terms is 15:7, then S₁₅ − S₅ is equal to?

a. 690 b.    890 c.   790 d.    800

Solution: (c)

We know that in an A. P Formula for the sum of n terms is

${\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}\cdot [2\mathrm{a}+(\mathrm{n}-1)\cdot \mathrm{d}]$

$S_{10}=\frac{10}{2}\cdot [2a+(10-1)\cdot d]$

390 = 5·(2a + 9d)

390 = 10a + 45d

78 = 2a + 9d . . . (1)

Now we have given the ratio of the tenth and fifth terms as 15: 7,

Then, in an A.P., the nth term is given by a_n = a + (n−1).d

So, the Tenth term a₁₀ = a + (10−1).d

a₁₀ = a + 9d

And the Fifth term is a₅ = a + (5−1).d

a₅ = a + 4d

We know the ratio as follows:

a+$\frac{9d}{a}$+4 d=$\frac{15}{7}$

7(a + 9d) = 15(a + 4d)

7a + 63d = 15a + 60d

63d – 60d = 15a –7a

3d = 8a . . . (2)

Solving Eq (1) and Eq (2) 

We get a =3, and d = 8

Now, we can find S₁₅ and S₅ by using the formula of the Sum of an AP

So, S₁₅ = 885 and S₅ = 95

So, S₁₅−S₅ = 885 – 95 = 790

Therefore, the correct answer is 790, Option (c).

2. Let the 2^nd, 8^th, and 44^th terms of a non-constant A. P. be, respectively, the 1^st, 2^nd, and 3^rd terms of a G. P. If the first term of the A. P. is 1, then the sum of its first 20 terms is equal to

a. 960 b. 980 c. 990 d. 970

Solution: (d)

From the given question the 2^nd, 8^th and 44^th terms of an A.P is 1 + d, 1 + 7d, and 1 + 43d.  

If these terms are in G. P, then,

(1 + 7d)² = (1 + d) (1 + 43d )

1 + 49d² + 14d = 1 + 44d + 43d²

6d² – 30d = 0

d = 5

Now, using the Formula of the Sum of the first n initial terms of an A.P

${\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}\cdot [2\mathrm{a}+(\mathrm{n}-1)\cdot d]$

${\mathrm{S}}_{20}=\frac{20}{2}\cdot [2.1+(20-1)\cdot 5]$

S₂₀ = 10[2 + 95]

S₂₀ =970

So, the correct answer is (d), which is 970.

6.0Sequence and Series Jee Advanced Question

Question 1: If the sum of the initial n terms of an Arithmetic Progression is cn², then the sum of the squares of these n terms is

1. $\frac{\mathrm{n}\left(4{\mathrm{n}}^2-1\right){\mathrm{c}}^2}{6}$
2. $\frac{\mathrm{n}\left(4{\mathrm{n}}^2+1\right){\mathrm{c}}^2}{3}$
3. $\frac{\mathrm{n}\left(4{\mathrm{n}}^2-1\right){\mathrm{c}}^2}{3}$
4. $\frac{\mathrm{n}\left(4{\mathrm{n}}^2+1\right){\mathrm{c}}^2}{6}$

Solution: (c) 

Let the sum of the first n terms of an A.P be S_n.

Given S_n = cn²

S_n–1 = c (n – 1)²

= cn² – cn² + 2cn –c

= c (2n –1)

Sum of squares = ΣTn²

= c²[4Σn² + Σ1 – 4 Σn]

=$\left[\frac{4{\mathrm{n}}^2\mathrm{n}(\mathrm{n}+1)(2\mathrm{n}+1)}{6}\right]+n{c}^2-\left[2{\mathrm{c}}^2\mathrm{n}(\mathrm{n}+1)\right]$

=$\frac{n{c}^2\left(4n^2+6n+2+3-6n-6\right)}{3}$

=$\frac{n{c}^2\left(4n^2-1\right)}{3}$

=$\frac{n\left(4n^2-1\right)c^2}{3}$

Hence, option(c) is the answer.

7.0Sequence and Series Solved Problem

Example 1: If a, b, c, d, and p are distinct real numbers such that the inequality (a² + b² + c²) p² – 2p(ab + bc + cd) + (b² + c² + d²) ≤ 0, then a, b, c, d are in

a. A.P b.    G.P c.    H.P d.    None of these

Solution: (b)

Here, the given condition (a² + b² + c²)p² – 2p(ab + bc + cd) + (b² + c² +d²) ≤ 0

This implies (ap – b)² + (bp – c)² + (cp – d)² ≤ 0

Therefore, a square cannot be negative

Therefore, ap –b = 0, bp – c = 0 implies $\mathrm{p}=\frac{\mathrm{b}}{\mathrm{a}}=\frac{\mathrm{c}}{\mathrm{b}}=\frac{\mathrm{d}}{\mathrm{c}}$ implies a, b, c, and dare in G.P.

Example 2: Let a, b, and c be positive integers s.t $\frac{b}{a}$ is an integer. If a, b, and c are in G. P and the arithmetic mean of a, b, and c is b + 2, then the value of $\frac{\left(a^2+a-14\right)}{(a+1)}$ is

Solution:

Given that a, b, and c are in geometric progression.

Let a, b, c be a, ar, ar² (where r is the common ratio)

Given $\frac{(a+b+c)}{3}$=b+2

Implies (a + ar + ar²) = 3(ar) + 6(since b = ar)

Implies ar² – 2ar + a = 6

⇒ a(r² – 2r + 1) = 6

⇒ a(r – 1)² = 6

⇒ (r – 1)² = $\frac{6}{a}$

$\frac{6}{a}$ should be a perfect square.

So, the possible value for a is 6.

$\frac{\left({\mathrm{a}}^2+\mathrm{a}-14\right)}{(\mathrm{a}+1)}=\frac{(36+6-14)}{7}=\frac{28}{7}=4$

Hence the value of $\frac{\left(a^2+a-14\right)}{(a+1)}$ is 4

8.0Sample Question on Sequence and Series

Q. What is the sum of an infinite geometric series?

Ans: The sum of infinite geometric series can be found using the formula $S=\frac{a_1}{1-r}$,  where the first term is a₁, and the common ratio between consecutive terms is r.