Arithmetic Sequences

Grade 4

What makes a sequence arithmetic

An arithmetic sequence (or arithmetic progression) is a list of numbers where the difference between consecutive terms is always the same. That fixed difference is called the common difference, d.

Examples: 3, 7, 11, 15 … (d = 4). 20, 17, 14, 11 … (d = −3).

The nth term formula

nth term = a + (n − 1)d

Where a is the first term and d is the common difference. This formula gives the value of any term in the sequence without listing them all out.

Worked example

EXAMPLE

Find the nth term of: 5, 8, 11, 14 … and use it to find the 50th term.

First term a = 5. Common difference d = 3. nth term = 5 + (n − 1)(3) = 5 + 3n − 3 = 3n + 2. For the 50th term: 3(50) + 2 = 152.

Is a value in the sequence?

Set the nth term equal to the value and solve for n. If n is a positive integer, the value is in the sequence. If n is a decimal or negative, it is not.

To find d from the sequence, subtract any term from the next one. Always check your formula gives the right first term — substitute n = 1.

Try this one

Here are the first five terms of an arithmetic sequence:

\( 7, \quad 11, \quad 15, \quad 19, \quad 23 \)

Find an expression for the nth term of this sequence. [2]

Nova's hint:

Finding the nth term of an arithmetic sequence involves identifying the common difference and first term.

Ask yourself: 'What is the common difference, and how does the first term relate to the formula \( an + b \)?'

For example, a sequence 5, 8, 11, ... has common difference 3, giving \( 3n + 2 \) as its nth term.

Try to Solve it with Nova?