Sequence and Series Formula

A sequence is an ordered list of numbers. The numbers in the list are the terms of the sequence. A series is the addition of all the terms of a sequence. Sequence and series are similar to sets but the difference between them is in a sequence, individual terms can occur repeatedly in various positions. The length of a sequence is equal to the number of terms, which can be either finite or infinite. Let us start learning Sequence and series formula.

Sequence and Series Formula

What are Sequence and Series?

A sequence is an ordered list of numbers. The numbers in the list are the terms of the sequence. The terms of a sequence usually name as a_i  or a_n, with the subscripted letter i or n being the index. So, the second term of a sequence might be named a₂, and a₁₂ would be the twelfth term.

A series termed as the sum of all the terms in a sequence. However, there has to be a definite relationship between all the terms of the sequence.

S_N = a₁+a₂+a₃ + .. + a_n

Types of Sequence and Series

Sequences: A finite sequence stops at the end of the list of numbers like a₁, a₂, a₃, a₄, a₅, a₆……a_n. whereas, an infinite sequence is never-ending i.e. a₁, a₂, a₃, a₄, a₅, a₆……a_n…..

Series: In a finite series, a finite number of terms are written like a₁ + a₂ + a₃ + a₄ + a₅ + a_{6 +} ……a_n. In case of an infinite series, the number of elements are not finite i.e. a₁ + a₂ + a₃ + a₄ + a₅ + a_{6 +} ……a_n +_…..

Some Common Sequences

Arithmetic Sequences:

A sequence in which every term is obtained by adding or subtraction a definite number to the preceding number is an arithmetic sequence.

Geometric Sequences:

A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence.

Harmonic Sequences:

If the reciprocals of all the elements of the sequence form an arithmetic sequence then the series of numbers is said to be in a harmonic sequence.

Fibonacci Numbers:

Fibonacci numbers form a sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Sequence is defined as, F₀ = 0 and F₁ = 1 and F_n = F_n-1 + F_n-2

Sequence and Series Formulas

The sequence of A.P: The n^th term a_n of the Arithmetic Progression (A.P) a, a+d, a+2d,…a, a+d, a+2d,… is given by

a_n=a+(n–1)d

Where,

a	First-term
d	Common difference
n	Position of the term
l	Last term

Arithmetic Mean: The arithmetic mean between a and b is given by A.M=\(\frac{a+b}{2}\)

The sequence of G.P: The nth term a_n of the geometric progression a, ar, ar², ar³,…,  is a_n=arⁿ–1an=ar^n–1

The geometric mean between a and b is G.M= ±\sqrt{ab}     

Sequence of H.P: The n^th term a_n of the harmonic progression is a_n= \( \frac{1}{a+(n–1)d} \)

The harmonic mean between a and b is H.M=\(\frac{2ab}{a+b}\)

Series of A.P: If S_n denotes the sum up to n terms of A.P. a, a+d, a+2d,…a, a+d, a+2d,… then

S_n = \(\frac{n}{2}(a+l),\)

S_n = \( \frac{n}{2} [2a+(n–1)d] \)

The sum of n A.M between a and b is  A.M = \(\frac{n(a+b)}{2}\)

Series of G.P: If S_n denotes the sum up to n terms of G.P is S_n=\(\frac{a(1–rn)}{1–r}\); r≠1 and l=arⁿ

The sum S of infinite geometric series is S=\(\frac{a}{1–r}; \)

Solved Examples

Question 1: If 1, 3, 5, 7, 9…… is a sequence, Find Common difference, nth term, 21st term

Solution: Given sequence is, 1, 3, 5, 7, 9……

1. a) common difference d = 3 – 1 = 2
2. b) The nth term of the arithmetic sequence is denoted by the term T_nand is given by T_n = a + (n-1) d,
3. c) 21^st term as:  T₂₁= 1 + (21-1)2 = 1+40 = 41.