Common Mistakes

Frequent Errors

1. Connecting sequence points into a continuous curve. 2. Using ratios to "check" an arithmetic sequence (or differences for geometric). 3. Writing $a_n=a_0 d n$ or $a_n=a_0+d^n$. 4. Writing $g_n=g_0+rn$ or $g_n=g_0 r+n$. 5. Off-by-one errors with $n-k$ in shift forms. 6. Dropping the sign when $r<0$. 7. Saying geometric sequences increase by the same amount each step. 8. Confusing term number with the term value.

Why These Happen

Students import continuous function graphing habits and mix additive/multiplicative templates from Unit 1 modeling.

How to Fix Them

• Always show the $d$ or $r$ calculation explicitly.
• Template cards: $a_0+dn$, $a_k+d(n-k)$, $g_0 r^n$, $g_k r^{n-k}$.
• Plot 4 points with open emphasis: "no connecting."
• Compare first-step vs later-step growth amounts for geo vs arith.

Avoidance Scripts

• "Add same $d$ = arithmetic; multiply same $r$ = geometric."
• "Sequences are dots."
• "Geometric growth amounts change; arithmetic growth amounts do not."

Quick Self-Check

• Did I verify $d$ or $r$ from successive terms?
• Does my formula reproduce a known term?
• Is my graph discrete?