I Know; What Do You Know?

Learning Focus

Identify whether a sequence is arithmetic, geometric, or neither, and write a rule when possible.

Big idea

Test a sequence in two steps:

1. Compute differences $f(n+1)-f(n)$. All equal? $\Rightarrow$ arithmetic.
2. Compute ratios $\dfrac{f(n+1)}{f(n)}$. All equal? $\Rightarrow$ geometric.

If neither is constant, the sequence is neither (for example, square numbers).

Worked example

Classify $1,4,9,16,\dots$

• Differences: $3,5,7$ — not constant
• Ratios: $4,\ 2.25,\ 1.78\dots$ — not constant
• So the sequence is neither.

Classify $20,17,14,11$: differences are all $-3$, so it is arithmetic with $f(n)=20-3(n-1)=23-3n$.

Takeaways

• Check differences first, then ratios.
• Check all consecutive pairs, not just the first two.
• Some sequences are neither; that's okay.