Geometric Meanies

Learning Focus

Find missing terms in a geometric sequence (geometric means) when the endpoints are known.

Big idea

With $k$ missing terms between a first and last term there are $k+1$ equal multiplications by $r$:

$r^{\,k+1}=\frac{\text{last}}{\text{first}}$

Solve for $r$, then multiply repeatedly. Be careful: an even number of steps can allow both a positive and negative $r$. Unless told otherwise, use the positive ratio.

Worked example

Insert 3 geometric means between $3$ and $48$.

• Steps: $3+1=4$, so $r^4=\dfrac{48}{3}=16$
• $r=2$ (or $-2$)
• Sequence: $3,\ 6,\ 12,\ 24,\ 48$

Check: multiplying by 2 four times gives $3\cdot16=48$ ✓.

Takeaways

• Geometric means use a common ratio, not a common difference.
• $k$ missing terms means the ratio is applied $k+1$ times.
• Even number of steps can produce $\pm r$.