Scott’s Muscle March

Learning Focus

Build a quadratic model from a cumulative total of an arithmetic pattern.

Big idea

Scott does 1 push-up on day 1, 2 on day 2, 3 on day 3, and so on. The daily amount grows by a constant 1, but the running total is a sum of those amounts:

$T(n)=1+2+3+\cdots+n=\frac{n(n+1)}{2}=\frac{1}{2}n^2+\frac{1}{2}n.$

Totals: $1,3,6,10,15,\dots$ First differences: $2,3,4,5,\dots$ Second differences: constant $1$. A constant second difference signals a quadratic, and the leading coefficient is half of it: $a=\frac{1}{2}$.

Worked example

Total push-ups after 8 days:

$T(8)=\frac{8\cdot 9}{2}=36.$

Takeaways

• Adding an arithmetic sequence term by term gives a quadratic total.
• Constant second difference $=2a$.
• A closed form like $\frac{n(n+1)}{2}$ replaces repeated addition.