Don’t Break the Chain

Learning Focus

Write sequences both recursively (using the previous term) and explicitly (using the term number), and use function notation $f(n)$.

Big idea

A sequence is a function whose input $n$ is the term number and whose output $f(n)$ is the term.

• Recursive rule: a starting value plus a rule linking $f(n)$ to $f(n-1)$.
  • Arithmetic: $f(n)=f(n-1)+d$
  • Geometric: $f(n)=r\cdot f(n-1)$
• Explicit rule: computes any term directly.
  • Arithmetic: $f(n)=f(1)+d(n-1)$
  • Geometric: $f(n)=f(1)\cdot r^{\,n-1}$

Recursive rules "keep the chain going" but you must know every earlier term. Explicit rules let you jump straight to term $n$.

Worked example

Sequence: $3,7,11,15,\dots$

• Recursive: $f(1)=3$, $f(n)=f(n-1)+4$
• Explicit: $f(n)=3+4(n-1)=4n-1$
• Check: $f(4)=4(4)-1=15$ ✓
• Jump ahead: $f(20)=4(20)-1=79$

Now try $2,6,18,54,\dots$: each term is $3$ times the last, so $f(1)=2$, $f(n)=3f(n-1)$, and explicitly $f(n)=2\cdot3^{\,n-1}$.

Takeaways

• A recursive rule needs a starting value and a rule that uses the previous term.
• Explicit rules are best for finding far-away terms.
• $f(n)$ is read "the $n$th term," not "$f$ times $n$."