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

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?