Arithmetic Sequence – Definition and Components

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”

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

1, 4, 7, 10, 13, 16, 19, …….

70, 62, 54, 48, 40, ……………

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

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) + a₁

d = 4

n = 98

a₁ = -4

a₉₈ = 4 × (98 – 1) + a₁

a₉₈ = 4 × (97) + -4

a₉₈ = 388 + -4

a₉₈ = 384

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