Introduction

What is a Sequence?

A set of numbers arranged according to some definite rule is called a sequence.
A sequence containing a finite number of terms is called a Finite Sequence.
A sequence that is not a finite sequence is called an Infinite Sequence.
Examples:

1. 3, 6, 9, 12, 15, 18 is a finite sequence. Here the successive terms are obtained by adding 3 to the previous term (rule).
2. 1, 2, 4, 8, 16, ........ is an infinite sequence. Here the successive terms are obtained by multiplying 2 by the previous term (rule).
3. 25, 20, 15, 10, ....... is an infinite sequence. Here the successive terms are obtained by subtracting 5 from the previous term (rule).
4. $3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \frac{3}{16}$ is a finite sequence. Here the successive terms are obtained by dividing the previous term by 2 (rule).
5. 2, 3, 5, 7, 11, 13, 17, ... is an infinite sequence of prime numbers.

What is an Arithmetic Sequence?

An arithmetic sequence or arithmetic progression is a special type of sequence in which the difference between any two consecutive terms is always the same (constant).
For example, the sequence 5, 10, 15, 20 ... is arithmetic because the difference between the consecutive terms is always five.
The sequence 7, 9, 11, 12 ... is not arithmetic because the difference between the consecutive terms is not the same throughout.
In general, an arithmetic sequence is expressed as: $a_{1}, (a_{1} + d), (a_{1} + 2 d), (a_{1} + 3 d), . . . . . . .$ , where $a_{1} =$ first term and $d =$ common difference.
The formula to find $n^{t h}$ term of an arithmetic sequence is $a_{n} = a_{1} + (n - 1) \cdot d$ .

What is an Arithmetic Series?

It is the sum of the terms of an arithmetic sequence.
The formula to find the sum of the first ' $n$ ' terms of an arithmetic sequence is $S_{n} = \frac{n}{2} [a_{1} + l] = \frac{n}{2} [2 a_{1} + (n - 1) d]$ , where $l =$ the last term.

Examples

Example 1: Is $1, (1 + \sqrt{3}), (1 + 2 \sqrt{3}), (1 + 3 \sqrt{3}), (1 + 4 \sqrt{3}), . . . . . . .$ an arithmetic sequence?

Solution: Since the common difference $(d = \sqrt{3})$ is same throughout, the given sequence is an arithmetic sequence.

Example 2: Find the common difference in the arithmetic sequence $5, \frac{9}{2}, 4, \frac{7}{2}, 3, . . . . .$

Solution: Common difference of an arithmetic sequence is always constant and can be found by taking the difference between any two successive terms. Here $d = (\frac{9}{2} - 5) = - \frac{1}{2}$ .

Example 3: Find the $5^{t h}$ term of an arithmetic sequence whose first term is $- 7$ and the common difference is $3$ .

Solution: It is given that $a_{1} = - 7$ and $d = 3$ .

We know that $a_{n} = a_{1} + (n - 1) \cdot d$

So, $a_{5} = a_{1} + (5 - 1) \cdot d = a_{1} + 4 d = (- 7) + 4 \times 3 = - 7 + 12 = 5$

Example 4: Find the $11^{t h}$ term from the end, of the arithmetic sequence $1, 5, 9, 13, 17, . . . . . . 69$ .

Solution: Here $a_{1} = 1$ and $d = (5 - 1) = 4$ .

We know that $m^{t h}$ term of an arithmetic sequence from the end is given by $a_{m} = l - (n - 1) \cdot d$

So, $a_{11} = 69 - (11 - 1) \times 4 = 69 - 40 = 29$

Example 5: Find the sum of 10 terms of an arithmetic sequence $10, 12, 14, 16, . . . . .$

Solution: Here $a_{1} = 10$ and $d = 2$ .

We know that sum of ' $n$ ' term of an arithmetic sequence is given by $S_{n} = \frac{n}{2} [2 a_{1} + (n - 1) \cdot d]$ .

So, $S_{10} = \frac{10}{2} \times [2 \times 10 + (10 - 1) \times 2] = 5 \times [20 + 18] = 5 \times 38 = 190$

Cheat Sheet

$n^{t h}$ term of an arithmetic sequence from the beginning: $a_{n} = a_{1} + (n - 1) \cdot d$
$m^{t h}$ term of an arithmetic sequence from the end: $a_{m} = [l - (n - 1) \cdot d]$
Sum of first ' $n$ ' terms of an arithmetic sequence: $S_{n} = \frac{n}{2} [a_{1} + a_{n}]$
Sum of first ' $n$ ' terms of an arithmetic sequence: $S_{n} = \frac{n}{2} [2 a_{1} + (n - 1) \cdot d]$
If there are only ' $n$ ' terms in an arithmetic sequence, then the $n^{t h}$ term is called the last term and is represented by the letter ' $l$ '.

Blunder Areas

It is not necessary that the terms of a sequence always follow a certain pattern or are described by some explicit formula for the $n^{t h}$ term.

Abhishek Tiwari
10 Comments
57 Likes

Algebra 2 - Arithmetic Sequences & Series