# Arithmetic Sequence – Definition and Components

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An arithmetic sequence is a sequence the place every time period is discovered by including or subtracting the identical worth from one time period to the subsequent. This worth that’s added or subtracted is known as “widespread sum” or “widespread distinction”

If the widespread distinction is constructive, the phrases of the sequence will improve in worth.

If the widespread distinction is unfavourable, the phrases of the sequence will lower in worth.

For instance, the next two sequences are examples of arithmetic sequences.

1, 4, 7, 10, 13, 16, 19, …….

70, 62, 54, 48, 40, ……………

Trying fastidiously at 1, 4, 7, 10, 13, 16, 19, ……., helps us to make the next remark:

As you possibly can see, every time period is discovered by including 3, a standard sum, to the earlier time period.

Trying fastidiously at 70, 62, 54, 46, 38, ……., helps us to make the next remark:

This time, to seek out every time period, we subtract 8, a standard distinction from the earlier time period.

## Many arithmetic sequences can me modeled with an algebraic expression

Here’s a trick or “**recipe per se**” to shortly get an algebraic expression!

**1)** Allow us to attempt to mannequin 1, 4, 7, 10, 13, 16, 19, …….

Let n signify any time period quantity within the sequence. The quantity we add to every time period is 3.

The quantity that comes proper earlier than 1 within the sequence is -2.

We are able to due to this fact mannequin the sequence with this algebra expression: 3 × n + -2.

**Test to see if the algebra expression works:**

- When n = 1, which represents the primary time period, we get 3 × 1 + -2 = 3 + -2 = 1

- When n = 2, which represents the second time period, we get 3 × 2 + -2 = 6 + -2 = 4

- When n = 3, which represents the third time period, we get 3 × 3 + -2 = 9 + -2 = 7

**The algebraic expression works!**

**2)** Allow us to attempt to mannequin 70, 62, 54, 46, 38, ……………

Let n signify any time period quantity within the sequence. The quantity we subtract to every time period is -8.

The quantity that comes proper earlier than 70 within the sequence is 78.

We are able to due to this fact mannequin the sequence with this algebraic expression: -8 × n + 78.

**Test to see if the algebra expression works:**

- When n = 1, which represents the primary time period, we get -8 × 1 + 78 = -8 + 78 = 70

- When n = 2, which represents the second time period, we get -8 × 2 + 78 = -16 + 78 = 62

- When n = 3, which represents the second time period, we get -8 × 3 + 78 = -24 + 78 = 54

**Once more, the algebraic expression works!**

## Arithmetic sequence method

The way in which that we modeled the arithmetic sequences above with algebraic expressions is a shortcut. We’ll now search for the arithmetic sequence method utilizing the algebraic expressions.

**1)**

3 × n + -2 is the algebraic expression for 1, 4, 7, 10, 13, 16, 19, …….

Allow us to attempt to rewrite 3 × n + -2 by making the first time period seem within the expression.

3 × n + -2 = 3 × n + -3 + 1 (since -2 = -3 + 1)

3 × n + -2 = 3 × (n – 1) + 1

3 is the quantity we add to every time period

1 is the primary time period

n is the variety of phrases

**2)**

-8 × n + 78 is the algebraic expression for 70, 62, 54, 46, 38, ……………

Allow us to attempt to rewrite -8 × n + 78 by making the first time period seem within the expression.

-8 × n + 78 = -8 × n + 8 + 70 (since 78 = 8 + 70)

-8 × n + 78 = -8 × (n – 1) + 70

-8 is the quantity we add to every time period

70 is the primary time period

n is the variety of phrases

**Normally, **

Let d be the quantity we add every time or the widespread distinction.

Let a_{1} be the primary time period

Let n be the variety of phrases

Let a_{n} be the nth time period.

Then, a_{n} = d × (n – 1) + a_{1}

## A few workouts about arithmetic sequences

Are the given sequences arithmetic? If that’s the case, discover the 98th time period.

**a.** 2, 6, 9, 11, ….

**b.** -4, 0, 4, 8, 12, ….

2, 6, 9, 11, …. isn’t an arithmetic sequence for the reason that quantity we add to every time period isn’t all the time the identical.

-4, 0, 4, 8, 12, …. is an arithmetic sequence for the reason that quantity we add to every time period is all the time the identical.

a_{n} = d × (n – 1) + a_{1}

d = 4

n = 98

a_{1} = -4

a_{98} = 4 × (98 – 1) + a_{1}

a_{98} = 4 × (97) + -4

a_{98} = 388 + -4

a_{98} = 384

## Take the arithmetic sequence quiz under to test your understanding of this lesson.

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