Sequences and Series

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 a_n​, 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 + . . .

There are many sequences in mathematics, but mainly three are asked in school exams, these are arithmetic, geometric, and harmonic:

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, and 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)

➣ Note: Besides these, another commonly known sequence is the Fibonacci Sequence.

Formulas for Sequence and Series

For arithmetic, geometric, and harmonic sequences, there are various formulas to calculate the n^th 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. These differences are listed in the following table:

➣ Read in detail - [ Sequence vs Series ]

Convergence and Divergence of Series

Given a sequence {a_n}, 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 S_N = a₁ + a₂ + ⋯ + a_N​ 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 S_N does not approach a finite limit as N→∞. In other words, if S_N​ either grows without bound or oscillates as N → ∞, the series diverges.

Read More about Convergence Tests.

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

Read More,

• Special Series in Maths
• Laurent Series
• Power Series

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 = a₁ + (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: a_n = 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:

a_n = 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: a_n = a₁ + (n − 1)⋅d, If the first term a₁ = 3 and the common difference d = 5, what is the 15th term of the sequence?