Sequences and Series (original) (raw)

Last Updated : 15 Oct, 2025

A **sequence is an ordered list of numbers following a specific rule. Each number in a sequence is called a "term." The order in which terms are arranged is crucial, as each term has a specific position, often denoted as an​, where n indicates the position in the sequence.

**For example:

• 2, 5, 8, 11, 14, . . . [Here, each term is 3 more than the previous term]
• 3, 6, 12, 24, 48, . . . [Here, each term is 2 times of the preceding term]
• 0, 1, 1, 2, 3, 5, 8, 13, 21, . . . [Here, each term is sum of two preceding terms]

A **series is the sum of the terms of a sequence. If we have a sequence a1, a2, a3, . . . the series associated with it is:

S = a1 + a2 + a3 + . . .

**Real-life example of a series: Saving money with a fixed deposit

Suppose you save ₹1,000 every month in a bank account that gives interest.
The total amount after a year is the sum of 12 deposits plus interest — that’s an arithmetic or geometric series, depending on how interest is applied.

**Arithmetic Sequence

An **arithmetic sequence (or arithmetic progression) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the **common difference (denoted as d).

**For example:

• 2, 5, 8, 11, 14, . . . (first term = 2 and common difference = 3)
• 10, 7, 4, 1, −2, . . . (first term = 10 and common difference = -3)
• 1, 2.5, 4, 5.5, 7, . . . (first term = 1 and common difference = 1.5)

The sequence in which each consecutive term has a common difference, and this difference could be positive, negative, or even zero, is known as an arithmetic sequence.

**Geometric Sequence

A **geometric sequence (or geometric progression) is a sequence of numbers in which the ratio between consecutive terms is constant. This ratio is known as the common ratio (denoted as r).

**For example:

• 3, 6, 12, 24, 48, . . . (first term = 3 and common ratio = 2)
• 1, 3, 9, 27, 81, . . . (first term = 1 and common ratio = 3)
• 16, 8, 4, 2, 1, . . . (first term = 16 and common ratio = 1/2)
• 5, −10, 20, −40, 80, . . . (first term = 5 and common ratio = -2)

**Harmonic Sequence

A **harmonic sequence (or harmonic progression) is a sequence of numbers where the reciprocals of the terms form an arithmetic sequence. In other words, if the sequence is a1, a2, a3, . . . , then the sequence of reciprocals 1/a1, 1/a2, 1/a3, . . . is an arithmetic sequence.

**For example:

• 1, 1/2​, 1/3​, 1/4​, 1/5​, . . . (as 1, 2, 3, 4, 5, . . . is arithmetic sequence)
• 3, 3/2, 1, 3/4, 3/5, . . . (1/3, 2/3, 3/3, 4/3, 5/3, . . . is arithmetic sequence)

Formulas for Sequence and Series

For arithmetic, geometric, and harmonic sequences, there are various formulas to calculate the nth term or the sum of the sequence. These formulas are:

Sequences vs Series

Sequence and series are often used interchangeably by many, but there is a very clear difference between them.

Convergence and Divergence of Series

Given a sequence {an}, the series is written as:

\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \dots

• A series \sum_{n=1}^{\infty} a_n​ **converges if the sequence of partial sums SN = a1 + a2 + ⋯ + aN​ approaches a finite limit as N → ∞:

\lim_{N \to \infty} S_N = S

Where S is a finite number. In this case, the series is said to have the sum S.

• A series \sum_{n=1}^{\infty} a_n​ **diverges if the sequence of partial sums SN does not approach a finite limit as N→∞. In other words, if SN​ either grows without bound or oscillates as N → ∞, the series diverges.

Special Series

Some special series are:

• **Arithmetic-Geometric Series (AGS) is a special type of series that combines both arithmetic and geometric sequences. It can be expressed in the form:

S = a + (a + d)x + (a + 2d)x2 + (a + 3d)x3 + . . .

• **Binomial Series is an infinite series that provides a way to expand expressions of the form (a + b)n, where n is any real number (not just a positive integer). The series is derived from the Binomial Theorem, which states that:

(a + b)^n = \sum_{k=0}^{\infty} \binom{n}{k} a^{n-k} b^k

• Taylor Series of a function f(x) about a point a is given by the formula:

f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \ldots

This can be expressed in summation notation as:

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n

• Maclaurin Series is a special case of the Taylor Series, where the expansion is around the point a = 0:

f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \ldots

In summation form, it is:

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n

**Also Check

• Special Series in Maths
• Laurent Series
• Power Series
• Fibonacci Sequence
• Convergence Tests

Solved Examples on Sequence and Series

**Question 1: Find the 10th term of the sequence: 4, 8, 12, 16, 20, ...

**Solution:

Use the formula for the nth term of an arithmetic sequence: **a n **= a 1 **+ (n − 1) ⋅ d

For n = 10

a10 = 4 + (10 − 1) ⋅ 4 = 4 + 36 = 40

**Answer: The 10th term is 40.

**Question 2: Find the sum of the first 6 terms of the sequence: 2, 6, 18, 54, 162, …

**Solution:

Use the sum formula for the first n terms of a geometric series:

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

For n = 6:

S_6 = 2 \cdot \dfrac{1 - 3^6}{1 - 3} = 2 \cdot \dfrac{1 - 729}{-2} = 2 \cdot \dfrac{-728}{-2}

**Answer: The sum of the first 6 terms is 728.

**Question 3: If the sequence is 1, 12, 13, …, find the sum of the first 5 terms of the harmonic series.

**Solution:

The harmonic sequence is \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}

S_5 = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} = 2.2833

**Answer: The sum of the first 5 terms is approximately 2.2833.

**Question 4: Calculate the sum of the first 15 terms of the sequence: −5, −2, 1, 4, 7, …

**Solution:

Use the sum formula for an arithmetic series:

S_n = \frac{n}{2} \cdot (2a_1 +(n-1) d)

For the first 15 terms:

• n = 15,
• a1 = −5,
• d = 3

Now, calculate the sum:

• S15 ​= 15/2​⋅(2⋅( −5) + (15 − 1)⋅3)
• S15​ = 15​/2 ⋅ ( -10 + 42)
• S15​ = 15​/2 ⋅ (32)
• S15​ = 15 ⋅ 16 = 240

**Answer: The sum of the first 15 terms is 240.

**Question 5: Find the sum of the infinite geometric series: \frac{5}{3} + \frac{5}{9} + \dots

**Solution:

Use the sum formula for an infinite geometric series (when ∣r∣<1): S_{\infty} = \frac{a}{1 - r}

where:

• a1 is the first term,
• r is the common ratio

The first term of the series is 5/3, and the common ratio is 1/3

Apply the formula for the infinite series:

The series converges if the absolute value of the common ratio ∣r∣<1|, which is true here because ∣r∣=1/3.

Now, applying the formula:

S_\infty = \dfrac{\frac{5}{3}}{1 - \dfrac{1}{3}} = \dfrac{\frac{5}{3}}{\dfrac{2}{3}} = \dfrac{5}{3} \times \dfrac{3}{2} = \dfrac{5}{2}

**Answer: The sum of the infinite series is 2.5

**Question 6: The nth term of a sequence is given by the formula: an = a1 + (n − 1) d. If the first term a1 = 10 and the common difference d = −2, what is the 8th term of the sequence?

**Solution:

Use the formula for the nth term of an arithmetic sequence:

an = a1 + (n − 1) ⋅ d

For n = 8:

a8 = 10 + (8 − 1)⋅(−2)
a8 = 10 + 7⋅(−2)
a8 = 10 − 14
= −4

**Answer: The 8th term is -4.

Practice Questions on Sequence and Series

1. Find the 12th term of the sequence: 5, 10, 15, 20, 25, ...
2. Find the sum of the first 8 terms of the sequence: 3, 9, 27, 81, 243, ...
3. If the sequence is 2, 6, 10, …, find the sum of the first 10 terms.
4. Calculate the sum of the first 20 terms of the sequence: 7, 11, 15, 19, 23, ...
5. Find the sum of the infinite geometric series: 1/4 + 1/16 + …
6. The nth term of a sequence is given by the formula: an = a1 + (n − 1)⋅d. If the first term a1 = 3 and the common difference d = 5, what is the 15th term of the sequence?