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

The \( n \)th term of a sequence is

\[ n^2 + 4n \]

One term of the sequence is equal to \( 165 \).

Find the position, \( n \), of this term.

[3]

Nova's hint:

Set the \( n \)th-term rule equal to the given value to form a quadratic equation, then solve it.

Ask yourself: 'once I have a quadratic equal to zero, can I factorise it or use the formula?' Only a positive whole-number solution counts as a valid position.

Try to Solve it with Nova?