Topic 2.1 Change in Arithmetic and Geometric Sequences: define arithmetic sequences by a constant common difference and geometric sequences by a constant common ratio, and write their explicit and recursive forms.

What this topic is asking

The College Board (Topic 2.1) wants you to distinguish two kinds of sequence by how their terms change. An arithmetic sequence adds a constant common difference $d$ each step; a geometric sequence multiplies by a constant common ratio $r$ each step. You must write both the explicit formula (the $n$th term directly) and the recursive formula (each term from the previous one).

Arithmetic sequences

The explicit form is a linear expression in $n$ (slope $d$, intercept $a_1 - d$), which is why arithmetic sequences are the discrete cousins of linear functions: equal steps in position produce equal changes in value.

Geometric sequences

The explicit form is an exponential expression in $n$, so geometric sequences are the discrete cousins of exponential functions: equal steps in position produce equal ratios, not equal differences.

Telling them apart

Given a list, test the change: subtract consecutive terms (constant difference means arithmetic) and divide consecutive terms (constant ratio means geometric). A sequence can be neither. The distinction mirrors the difference-versus-ratio tests for functions in Topics 1.3 and 2.2.

Why the indexing matters

The most common slip is the exponent and the multiplier of $(n - 1)$ rather than $n$. Both explicit forms count steps from the first term, and there are $n - 1$ steps to reach the $n$th term, which is why arithmetic uses $(n - 1)d$ and geometric uses $r^{\,n-1}$. Anchoring on "how many steps from term $1$ to term $n$" prevents the off-by-one errors that otherwise creep in, and it also explains why $g_1$ has the exponent $r^0 = 1$, leaving the first term unchanged.

A deeper connection worth noting is that arithmetic and geometric sequences are the discrete templates for the two function families of this unit. Plot an arithmetic sequence and the points lie on a line; plot a geometric sequence and they lie on an exponential curve. This is exactly why Topic 2.2 pairs linear with exponential change: the sequence behavior you classify here becomes the continuous-function behavior you model next, and recognizing the bridge makes the whole unit cohere.

Try this

Q1. Write the recursive formula for the geometric sequence $2, 6, 18, 54, \dots$. [1 point]

• Cue. Common ratio $r = 3$, so $g_n = 3 g_{n-1}$ with $g_1 = 2$.

Q2. Find the $10$th term of the geometric sequence with $g_1 = 4$ and $r = 2$. [1 point]

• Cue. $g_{10} = 4 \cdot 2^{9} = 4 \cdot 512 = 2048$.