Study Guide

Learning Objectives

• Express arithmetic sequences as functions of whole numbers; recognize discrete graphs (2.1.A.1).
• Identify the common difference $d$ as constant rate of change (2.1.A.2).
• Write $a_n=a_0+dn$ or $a_n=a_k+d(n-k)$ (2.1.A.3).
• Express geometric sequences as functions of whole numbers; identify common ratio $r$ (2.1.B.1).
• Write $g_n=g_0 r^n$ or $g_n=g_k r^{n-k}$ (2.1.B.2).
• Contrast equal-step arithmetic growth with successively larger geometric growth when increasing (2.1.B.3).

Suggested Practice Sequence

1. Classify: common difference vs common ratio from short lists. 2. Write $a_n$ from $a_0,d$; find $a_n$ values. 3. Write $a_n$ from a known $a_k$ and $d$. 4. Write $g_n$ from $g_0,r$; include $0<r<1$ and $r<0$. 5. Write $g_n$ from a known $g_k$ and $r$. 6. Contrast growth amounts for increasing arith vs geo sequences. 7. Context stories: seating rows, repeated interest-style multipliers (sequence form).

Classroom / Check-In Moves

• Two-column: differences vs ratios for the same list.
• Require "points only" when sketching.
• Cold-call the $(n-k)$ shift forms.

Misconception Watch-List

• Connecting sequence points into a continuous graph.
• Calling a geometric list arithmetic because terms are "going up."
• Using $a_n=a_0+d^n$ or $g_n=g_0+rn$.
• Indexing errors ($n$ vs $n-k$).

Vocabulary to Own

• sequence; whole-number input; discrete graph
• arithmetic sequence; common difference $d$
• geometric sequence; common ratio $r$
• general term $a_n$, $g_n$; initial value $a_0$, $g_0$
• proportional change vs constant rate of change

Materials Map

• Practice: classify, write formulas, find terms
• Word: contextual sequences
• Mixed / multi-*: contrast arith/geo and preview linear/exponential language lightly
• Review & practice test: timed fluency
• Interactive: quick checks