What is Sequence and Series: Formulas, Difference and Examples

The sequence is a collection of different objects arranged in such an orderly manner that it can be identified as the first, second, third member and so on.  A series definition states that it is the sum of all the terms present in a sequence. But there has to be a relationship between the terms and the sequence.

1. Sequence

When a sequence has a finite number of terms, it is called a Finite sequence.
E.g., the sequence of ancestors in a family is an excellent example of a finite sequence.

When a sequence has an infinite number of terms, it is called an infinite sequence.
E.g., a Fibonacci series is an excellent example of an infinite sequence.

2. Series

A mathematical series can be defined as infinite, or finite series based on their sequence. The series is always mentioned in short form, called sigma notation and denoted by the Greek letter ‘Sigma’.
E.g., if S1, S2, S3,…., Sn are a given sequence, then the series is defined as S1+ S2+ S3+ …. + Sn.

Let us consider some questions based on sequence and series formula. 

Q1. The first term of a GP is 1. The sum of the third and the fifth terms is 90. Find the common ratio of GP.

Solution. Let common ratio = r.

= a³ + a⁵ =90

= ar² + ar⁴ = 90

= r⁴ + r² -90 = 0

= r⁴ + 10r² -9r²- 90 =0

= r² - 9 = 0

= r = +3 or -3

Q2. If the sum of three numbers in AP is 24, and the product is 440, what are the numbers?

Solution. Let the three numbers be a-d, a, and a+d

Sum= (a-d)+ a + (a+1) = 24

      = 3a = 24

      = a = 8

Product= (a-d) * a * (a+d)= 440         

= d= 3

When d= +3, the numbers are 5, 8, and 11.

when d= =3, the numbers are 11, 8, and 5.

Q3. Find the sum of all numbers between 200 and 400, which are divisible by 7.

Solution. First terms, a= 203, Last term, an= 399, Difference= 7

an = a + (n-1)d

= 399= 203 + (n-1)*7

= n = 29

So, S₂₉ = (29/2)* (203 + 399)

S₂₉ = 8729

Let us now take a look at the types of sequence and series one by one. 

1. Types of Sequence

1. Arithmetic Progression: A sequence in which each term is formed by adding the same fixed number (called the common difference) to the preceding term. An arithmetic progression formula will be $a_{n+1}=a_n+d,n\in \mathbb{\mathbb{N}.}$Here, d is the common difference and a is the first term of AP. l is the last term of AP.
2. Geometric Progression: A sequence in which each term is formed by multiplying or dividing the same fixed number (called the common ratio) by the preceding term.
3. Harmonic Progression Sequence: A sequence whose terms are the reciprocals of consecutive positive integers, so each term equals $1/n$ for some integer 𝑛.
4. Fibonacci-Type (Recursive) Sequence: A sequence in which each term is produced from a fixed recurrence relation such as $F_n=F_{n-1}+F_{n-2}$ using two initial seed values.
5. Alternating Sequence: A sequence in which the sign of each term flips in a regular pattern, commonly written as $a_n={\left(-1\right)}^n{b}_n$ where $b_n$ is positive.
6. Monotone Sequence: A sequence that never reverses direction: it is entirely non-decreasing or entirely non-increasing from one term to the next.
7. Bounded Sequence: A sequence whose terms all stay within a single finite interval; there exist real numbers L and U such that $L\le a_n\le U$ for every 𝑛.
8. Random (Stochastic) Sequence: A sequence whose terms are random variables generated by some probability rule (e.g., i.i.d. draws or a Markov chain).

2. Types of Series

The following are different types of series:

• Finite Arithmetic Series  The sum of a fixed number of terms from an arithmetic sequence is $S_n=\frac{n}{2}(2a1+(n-1\left)d\right)$.
• Infinite Geometric Series  The sum of all terms of a geometric sequence converges when the common ratio’s absolute value is less than 1: $\underset{k=0}{\overset{\infty }{\sum }}a1r^k=\frac{a}{1}$, where  $\left|r\right|<1$.
• Harmonic Series  The infinite sum $\underset{n=1}{\overset{\infty }{\sum }}\frac{1}{n}$ diverges even though its individual terms approach zero.
• p-Series  An infinite series of the form $\underset{n=1}{\overset{\infty }{\sum }}\frac{1}{n^p}$ converges for  $p>1$ and diverges for  $p\le 1$.
• Alternating Series  An infinite series whose terms alternate in sign and decrease in absolute value, $\underset{n=1}{\overset{\infty }{\sum }}{-}^{n-1}b{}_n$, converges by the Leibniz criterion.
• Telescoping Series  A series in which many terms cancel when written in partial-fraction form, leaving only a small difference between the first and last surviving terms — e.g.  $\underset{n=1}{\overset{\infty }{\sum }}\frac{1}{n}-\frac{1}{n+1}=1$.
• Power Series  An infinite series expressed as  $\underset{n=0}{\overset{\infty }{\sum }}c{}_n{(x-a)}^n$, which converges inside a specific radius around  $x=a$.
• Taylor (or Maclaurin) Series  A power series whose coefficients come from the derivatives of a function, recreating that function near a chosen expansion point:  $f\left(x\right)=\underset{n=0}{\overset{\infty }{\sum }}\frac{f^{\left(n\right)}}{n^{!}}{(x-a)}^n$.
• Fourier Series  A representation of a periodic function as an infinite sum of sine and cosine terms with specific coefficients:  $\frac{a}{0}/2+\underset{n=1}{\overset{\infty }{\sum }}a{}_n\cos \left(nx\right)+b{}_n\sin \left(nx\right)$.
• Binomial Series  The expansion  $(1+x){}^k=\underset{n=0}{\overset{\infty }{\sum }}(kn)x^n$ valid for real (even fractional) exponents  $k$ when  $\left|x\right|<1$.
• Exponential Series  The universal expansion  $e^x=\underset{n=0}{\overset{\infty }{\sum }}\frac{x^n}{n^{!}}$, convergent for every real or complex value of  $x$.
• Conditionally Convergent Series  A series that converges when its terms carry their original signs but diverges when all terms are replaced by their absolute values, e.g.  $\underset{n=1}{\overset{\infty }{\sum }}\frac{{-}^{n+1}}{n}$ (the alternating harmonic series).
• Absolutely Convergent Series  A series that still converges after every term is replaced by its absolute value; absolute convergence guarantees stability under rearrangement, e.g.  $\underset{n=1}{\overset{\infty }{\sum }}\frac{1}{n^2}$ (the Basel series).

Let us understand the difference between sequence and series through the following table:

The following table has sequence and series formulas:

NEET and JEE Main aspirants must remember series and sequence formula by heart since questions based on these are frequently asked in competitive exams.