Arithmetic and Geometric Sequences

Patterns in Sequences

A sequence is an ordered list of numbers $a_1, a_2, a_3, \dots$. Two patterns are so common they have names and formulas.

An arithmetic sequence adds a fixed common difference $d$ each step: $a_n = a_1 + (n-1)d$. For $2, 5, 8, 11, \dots$, $d = 3$. The sum of the first $n$ terms (an arithmetic series) has a famous closed form: $$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}\big(2a_1 + (n-1)d\big).$$ The first form is the memorable one: the sum equals the number of terms times the average of the first and last. (This is the trick young Gauss used to add $1$ to $100$ instantly: $\frac{100}{2}(1 + 100) = 5050$.)

A geometric sequence multiplies by a fixed common ratio $r$ each step: $a_n = a_1 r^{n-1}$. For $3, 6, 12, 24, \dots$, $r = 2$. Its finite sum is $$S_n = a_1 \frac{r^n - 1}{r - 1} \quad (r \neq 1).$$ When $|r| < 1$, the infinite geometric series converges to $\frac{a_1}{1 - r}$ — infinitely many terms with a finite total, as with $1 + \tfrac12 + \tfrac14 + \dots = 2$.

The essential distinction is additive versus multiplicative growth. Arithmetic sequences change by a constant difference (linear growth — a straight line); geometric sequences change by a constant ratio (exponential growth — accelerating or decaying). Telling them apart is the first step in analyzing any pattern: check whether successive terms differ by a constant or are in a constant ratio.

Common pitfall: mixing up arithmetic (constant difference) with geometric (constant ratio) sequences, or their formulas. In an arithmetic sequence you add $d$; in a geometric sequence you multiply by $r$. So $2, 5, 8, 11$ is arithmetic ($+3$) while $2, 6, 18, 54$ is geometric ($\times 3$). Applying the arithmetic sum formula to a geometric sequence (or vice versa) gives nonsense — first identify which pattern you have.

Two bar rows: an arithmetic sequence with equal accent steps added (a straight-line top) beside a geometric sequence with each bar a constant multiple of the last (an accelerating curve) — additive versus multiplica.

$$a_n = a_1 + (n-1)d \;;\; a_n = a_1 r^{n-1}$$