Growing, Growing Dots

Learning Focus

Model a sequence with a common ratio using tables, graphs, and equations.

Big idea

A geometric sequence multiplies by a constant ratio $r$ each step.

• Recursive: $f(n)=r\cdot f(n-1)$
• Explicit: $f(n)=f(0)\cdot r^n$

Worked example

Start with $5$ dots and double each minute:

• $f(0)=5$, $r=2$
• Explicit: $f(t)=5\cdot 2^t$
• At $t=5$: $5\cdot 32=160$

Takeaways

• Common ratio $\Rightarrow$ geometric sequence / exponential family.
• Explicit form uses exponents; recursive form multiplies the previous term.