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# 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.  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:  , where ${a}_{1}=$first term and $d=$common difference.
• The formula to find ${n}^{th}$ term of an arithmetic sequence is ${a}_{n}={a}_{1}+\left(n-1\right)·d$.

# 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 ${S}_{n}=\frac{n}{2}\left[{a}_{1}+l\right]=\frac{n}{2}\left[2{a}_{1}+\left(n-1\right)d\right]$, where $l=$the last term.

## Examples

Example 1: Is  an arithmetic sequence?

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

Example 2: Find the common difference in the arithmetic sequence

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

Example 3: Find the ${5}^{th}$ 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 ${a}_{n}={a}_{1}+\left(n-1\right)·d$

So, ${a}_{5}={a}_{1}+\left(5-1\right)·d={a}_{1}+4d=\left(-7\right)+4×3=-7+12=5$

Example 4: Find the ${11}^{th}$ term from the end, of the arithmetic sequence .

Solution: Here ${a}_{1}=1$ and $d=\left(5-1\right)=4$.

We know that ${m}^{th}$ term of an arithmetic sequence from the end is given by ${a}_{m}=l-\left(n-1\right)·d$

So, ${a}_{11}=69-\left(11-1\right)×4=69-40=29$

Example 5: Find the sum of 10 terms of an arithmetic sequence

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

We know that sum of '$n$' term of an arithmetic sequence is given by ${S}_{n}=\frac{n}{2}\left[2{a}_{1}+\left(n-1\right)·d\right]$.

So, ${S}_{10}=\frac{10}{2}×\left[2×10+\left(10-1\right)×2\right]=5×\left[20+18\right]=5×38=190$

## Cheat Sheet

• ${n}^{th}$ term of an arithmetic sequence from the beginning: ${a}_{n}={a}_{1}+\left(n-1\right)·d$
• ${m}^{th}$ term of an arithmetic sequence from the end: ${a}_{m}=\left[l-\left(n-1\right)·d\right]$
• Sum of first '$n$' terms of an arithmetic sequence: ${S}_{n}=\frac{n}{2}\left[{a}_{1}+{a}_{n}\right]$
• Sum of first '$n$' terms of an arithmetic sequence: ${S}_{n}=\frac{n}{2}\left[2{a}_{1}+\left(n-1\right)·d\right]$
• If there are only '$n$' terms in an arithmetic sequence, then the ${n}^{th}$ 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 ${n}^{th}$ term.