Geometric and Special Sequences

Grade 5

Geometric sequences

In a geometric sequence, each term is multiplied by a fixed number called the common ratio, r. Example: 2, 6, 18, 54 … (r = 3). Or 80, 40, 20, 10 … (r = 1/2).

The nth term formula is ar^(n−1), where a is the first term.

Quadratic sequences

A quadratic sequence has a constant second difference. The differences between terms are not constant, but the differences of those differences are. The nth term contains an n² part.

If the second difference is 2k, then the nth term starts with kn². Find k, subtract kn² from each term, and find the arithmetic part from what remains.

EXAMPLE

Find the nth term of: 3, 8, 15, 24, 35 …

First differences: 5, 7, 9, 11 (increasing by 2). Second difference = 2, so k = 1: starts with n². Subtract n²: 3−1=2, 8−4=4, 15−9=6, 24−16=8. These go up by 2, giving the arithmetic part 2n. nth term = n² + 2n. Check: n=1: 1+2=3 ✓, n=2: 4+4=8 ✓. nth term = n² + 2n.

Fibonacci-type sequences

A Fibonacci-type sequence is formed by adding the two previous terms. Example: 1, 1, 2, 3, 5, 8, 13 … There is no simple nth term formula — you build it term by term.

Examiners sometimes give a sequence and ask you to identify the type before finding the nth term. Check differences first (arithmetic), then second differences (quadratic), then ratios (geometric).

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?