Hi there. In this guide, I cover the math topic of arithmetic sequences.

## Topics

- Basic Number Patterns
- The Arithmetic Sequence
- Examples Of Solving For Missing Values In Arithmetic Sequences

## Basic Number Patterns

Before getting into the definition of a arithmetic sequence I would like to share some basic number patterns. That is the number patterns you would have most likely seen as a kid or young student.

**Count By 2**

`2, 4, 6, 8, 10, 12, ..., 100`

**Count By 10 Starting at 7**

`7, 17, 27, 37, 47, ..., 177`

**Subtract 5 Every Time**

`28, 23, 18, 13, 8, 3, -2, -7, -12, ..., -42`

## The Arithmetic Sequence

In a more mathematical setting the arithmetic sequence is based on these basic number patterns where the increase or decrease is the same in getting the next number. Here is a formula for obtaining any number in an arithmetic sequence.

`t_n = t_1 + (n - 1)d`

Note that the term `t_1`

is the first number in the sequence.

The term is `t_n`

is the n-th term in the sequence. If `n = 10`

we have the tenth number or term in the arithmetic sequence.

The variable `n`

is the number of terms in the arithmetic sequence.

The letter `d`

represents how much the numbers go up or down in every step. A value of 2 for d would be counting up by 2 to get the next number as an example. A value of `-10`

for `d`

would be subtracting 10 to get the next number.

**Example One - Count By 8 Starting at 2**

`2, 10, 18, 26, 34, 42, 50, 58, 66`

The first term here is two with a `d`

value of eight. Fifty would be the seventh term in this sequence which you can represent as `t_7 = 50`

.

As the number of terms in this sequence is 9 the value for `n`

would be 9.

**Example Two - Subtract By 10 Starting at 59**

`59, 49, 39, 29, 19, 9, -1, -11, -21, -31`

With this arithmetic sequence we have:

- Ten values which would make
`n = 10`

- The first term as 59
- The final or tenth term as -31
- The common difference as negative ten

## Examples Of Solving For Missing Values In Arithmetic Sequences

The arithmetic sequence formula can be used to find certain numbers in a sequence, the first number in the sequence, how many numbers are there in the sequence or the common difference.

**Example One**

The first number in the sequence is 2 with a common difference of 5. What is the twentieth number in this arithmetic sequence?

You could start at 2 and count by five until you get number/term 20. That could take a bit of time. The mathematical way of doing this would be using the arithmetic sequence formula.

**Example Two - Find Number of Terms**

Start with the number one and the last number in the arithmetic sequence as 232. The common difference is `11`

. How many terms are there in this sequence?

The first number or `t_1`

is 1 and let `t_n = 232`

. Eleven would be the common difference as in `d = 11`

. Use the arithmetic sequence formula, substitute the known values and solve for `n`

.

**Example Three - Determine Starting Number**

An arithmetic sequence has a final number of `-108`

with a common difference of `-8`

. There are 16 numbers in this sequence. What is the first number here?

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