What Does It Mean?

Learning Focus

Find missing terms in an arithmetic sequence (arithmetic means) when the endpoints are known.

Big idea

If you know two terms of an arithmetic sequence with $k$ missing terms between them, there are $k+1$ equal steps from the first to the last.

$d=\frac{\text{last}-\text{first}}{k+1}$

Then add $d$ repeatedly to fill in the gaps. The inserted terms are called arithmetic means.

Worked example

Insert 3 arithmetic means between $5$ and $29$.

• There are $3+1=4$ steps.
• $d=\dfrac{29-5}{4}=6$
• Sequence: $5,\ 11,\ 17,\ 23,\ 29$

Check: each step adds 6 and the last term is 29 ✓.

Takeaways

• With $k$ missing terms there are $k+1$ steps.
• Divide the total change by the number of steps to get $d$.
• Always check that the endpoint matches.