What Comes Next? What Comes Later?

Learning Focus

Compare arithmetic growth (adding) with geometric growth (multiplying) for near and far terms.

Big idea

• Arithmetic: add a constant difference $d$. Explicit: $f(n)=f(1)+d(n-1)$. Graph points lie on a line.
• Geometric: multiply by a constant ratio $r$. Explicit: $f(n)=f(1)\cdot r^{\,n-1}$. Graph points curve upward (for $r>1$).

Early terms can look similar, but with $r>1$ a geometric sequence eventually outgrows any arithmetic sequence.

Worked example

Compare $5,8,11,\dots$ (add 3) with $5,10,20,\dots$ (double).

| $n$ | arithmetic | geometric | |---|---|---| | 1 | 5 | 5 | | 2 | 8 | 10 | | 3 | 11 | 20 | | 10 | $5+3(9)=32$ | $5\cdot2^9=2560$ |

Same start, very different "later."

Takeaways

• Constant differences mean arithmetic; constant ratios mean geometric.
• Use explicit formulas to answer "what comes later?"
• Growth by a factor greater than 1 eventually beats growth by adding.