# Arithmetic Sequences

## Key Questions

• An arithmetic sequence is a sequence (list of numbers) that has a common difference (a positive or negative constant) between the consecutive terms.

Here are some examples of arithmetic sequences:
1.)7, 14, 21, 28 because Common difference is 7.
2.) 48, 45, 42, 39 because it has a common difference of - 3.

The following are not examples of arithmetic sequences:

1.) 2,4,8,16 is not because the difference between first and second term is 2, but the difference between second and third term is 4, and the difference between third and fourth term is 8. No common difference so it is not an arithmetic sequence.

2.) 1, 4, 9, 16 is not because difference between first and second is 3, difference between second and third is 5, difference between third and fourth is 7. No common difference so it is not an arithmetic sequence.

3.) 2, 5, 7, 12 in not because difference between first and second is 3, difference between second and third is 2, difference between third and fourth is 5. No common difference so it is not an arithmetic sequence.

${a}_{n} = {a}_{1} + \left(n - 1\right) \cdot d$
$\textcolor{w h i t e}{\text{XXX}}$where ${a}_{1}$ is the first term and
$\textcolor{w h i t e}{\text{XXXXXXX}} d$ is the difference between a term and its previous term.

#### Explanation:

Examine the pattern:

${a}_{1}$
${a}_{\textcolor{b r o w n}{2}} = {a}_{1} + d = \textcolor{g r e e n}{{a}_{1} + 1 d}$
${a}_{\textcolor{b r o w n}{3}} = {a}_{2} + d = {a}_{1} + d + d = \textcolor{g r e e n}{{a}_{1} + 2 d}$
${a}_{\textcolor{b r o w n}{4}} = {a}_{3} + d = {a}_{1} + 2 d + \mathrm{dc} o l \mathmr{and} \left(g r e e n\right) \left(= {a}_{1} + 3 d\right)$
${a}_{\textcolor{b r o w n}{5}} = {a}_{4} + d = {a}_{1} + 3 d + d = \textcolor{g r e e n}{{a}_{1} + 4 d}$

To find out the common difference in an AP you can perform the following simple step.

#### Explanation:

Subtract the first term of the AP from the second term of the AP.

d = ${a}_{2}$ - ${a}_{1}$

where d = common difference
${a}_{2}$ = any term other than first term
${a}_{1}$ = previous term

For example;

In the AP
3 , 9 , 15 , 21 , 27 , 33

Taking ${a}_{1}$ = 3
Taking ${a}_{2}$ = 9

${a}_{2}$ - ${a}_{1}$ = 9 - 3 = 6

hence , common difference or d = 6

Thanks

I hope this helps

The even numbers, the odd numbers, etc

#### Explanation:

An arithmetic sequence is builded up adding a constant number (called difference) following this method

${a}_{1}$ is the first element of a arithmetic sequence, ${a}_{2}$ will be by definition ${a}_{2} = {a}_{1} + d$, ${a}_{3} = {a}_{2} + d$, and so on

Example1:

2,4,6,8,10,12,....is an arithmetic sequence because there is a constant difference between two consecutive elements (in this case 2)

Example 2:

3,13,23,33,43,53,.... is an arithmetic sequence because there is a constant difference between two consecutive elements (in this case 10)

Example 3:

$1 , - 2 , - 5 , - 8 , \ldots$ is another arithmetic sequence with difference $- 3$

Hope this help