Family Overview

In Plain Language

A sequence is a function from the whole numbers to the reals, so its graph is discrete points, not a connected curve.

• Arithmetic sequences change by a common difference $d$ (constant additive rate of change). Formulas: $a_n=a_0+dn$ or $a_n=a_k+d(n-k)$.
• Geometric sequences change by a common ratio $r$ (constant proportional change). Formulas: $g_n=g_0 r^n$ or $g_n=g_k r^{n-k}$.

When both are increasing, arithmetic sequences add the same amount each step, while geometric sequences ($r>1$) increase by a larger amount with each successive step.

Why It Matters

Unit 2 builds exponential thinking from geometric sequences and links arithmetic sequences to linear change. AP Precalculus asks students to write closed forms, find terms, and contrast the two growth styles.

How Families Can Help

• Ask: "Are we adding the same number, or multiplying by the same number?"
• Practice: next three terms, then write $a_n$ or $g_n$.
• Emphasize: plot points only--do not connect into a solid curve.

What Not to Do

• Do not treat sequence graphs as continuous lines/curves.
• Do not mix $d$ (add) with $r$ (multiply).
• Do not assume geometric growth adds a constant amount each step.
• Do not forget the shift form using a known $a_k$ or $g_k$.