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 + . . .
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.
For arithmetic, geometric, and harmonic sequences, there are various formulas to calculate the nth term or the sum of the sequence. These formulas are:
| Type | Formula | Description |
|---|
| nth term of an Arithmetic Sequence | an = a1 + (n − 1)d | nth term of an arithmetic sequence |
| Sum of Arithmetic Series | Sn = 2n(a1 + an) | Sum of the first n terms of an arithmetic series |
| nth term of Geometric Sequence | an = a ⋅ rn−1 | nth term of a geometric sequence |
| Sum of Geometric Series (Finite) | Sn = a(1 − rn)/(1 − r) | Sum of the first n terms of a geometric series |
| Sum of Geometric Series (Infinite) | S∞ = a/(1 − r) (For r < 1} | The sum of the infinite geometric series where r < 1. |
| Harmonic Series | Hn = ∑nk=1 (1/k) | Sum of the first n terms of the harmonic series |
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:
| Aspect | Sequence | Series |
|---|
| Definition | An ordered list of numbers, following a specific rule or pattern. | The sum of the terms of a sequence. |
| Notation | Typically denoted as an or {an}. | Typically denoted as Sn or ∑an. |
| Example | 1, 2, 3, 4, 5, . . . (Arithmetic sequence) | 1 + 2 + 3 + 4 + 5 + . . . (Sum of the sequence) |
| Main Focus | Focuses on the terms themselves. | Focuses on the sum of the terms. |
| Representation | Written as a list or a formula for the nth term. | Written using summation notation (∑). |
| Use Cases | Used to define patterns or behaviors in data sets. | Used to calculate totals, averages, or in calculus for convergence. |
| Convergence | Not applicable; it is a list of values. | Can converge to a limit (infinite series) or diverge. |
➣ Read in detail - [ Sequence vs Series ]
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.
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,
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: an = 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:
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
- Find the 12th term of the sequence: 5, 10, 15, 20, 25, ...
- Find the sum of the first 8 terms of the sequence: 3, 9, 27, 81, 243, ...
- If the sequence is 2, 6, 10, …, find the sum of the first 10 terms.
- Calculate the sum of the first 20 terms of the sequence: 7, 11, 15, 19, 23, ...
- Find the sum of the infinite geometric series: 1/4 + 1/16 + …
- 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?
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