Arithmetic and geometric sequences

Look at how consecutive terms are related. If each term is the previous one plus a constant d, the sequence is arithmetic: 3, 7, 11, 15 adds 4 each step. If each term is the previous one times a constant r, it is geometric: 3, 6, 12, 24 multiplies by 2 each step.

The test is mechanical: subtract consecutive terms, and if the differences match, it is arithmetic. Divide consecutive terms, and if the ratios match, it is geometric. A sequence like 2, 4, 8, 14 passes neither test. And do not decide from the first two terms alone: 2, 4 could continue 6, 8 (adding 2) or 8, 16 (doubling). Two terms never determine the pattern.

Both families have a closed form built from the starting term and the step rule:

1. Arithmetic: $a_n = a_0 + d \cdot n$. Start at $a_0$, take n additive steps of size d.
2. Geometric: $g_n = g_0 \cdot r^n$. Start at $g_0$, take n multiplicative steps of factor r.

The subscript is a step count from the starting term, and that is where points die. If your sequence is indexed from $a_1$ instead of $a_0$, the number of steps from the start to $a_n$ is $n - 1$, and the forms become $a_n = a_1 + d(n-1)$ and $g_n = g_1 \cdot r^{\,n-1}$. Neither version is more correct; what matters is that the exponent or multiplier counts steps taken, not the label on the term. Before using any formula, ask: how many steps separate the term I know from the term I want?

The same sequences can be written recursively: state the first term, then the rule for getting the next. For 3, 7, 11, 15: $a_0 = 3$, $a_{n+1} = a_n + 4$. For 3, 6, 12, 24: $g_0 = 3$, $g_{n+1} = 2g_n$.

A recursive rule answers "how do I take one step?" An explicit form answers "where am I after n steps?" Confusing them produces the classic error of writing $a_n = a_n + 4$ as if it were a formula, or answering a "find the 20th term" question with the step rule. To reach term 20 recursively you must take the step twenty times; the explicit form jumps there in one evaluation.

Arithmetic growth is a straight-line story: equal steps forever, so the terms of 3, 7, 11, ... sit on the line $y = 3 + 4n$. Geometric growth compounds: each step multiplies everything accumulated so far, so 3, 6, 12, ... sits on the curve $y = 3 \cdot 2^n$, and no arithmetic sequence outruns it for long.

This is the bridge to the rest of the unit: a linear function is an arithmetic sequence with the integers filled in, and an exponential function is a geometric sequence with the integers filled in. When a table, a story, or a bank account grows "by the same amount" it is the first family; "by the same percent" or "by the same factor" is the second. The words amount and factor are the whole diagnosis.