Arithmetic and Geometric Sequences

Arithmetic Sequences

A sequence is an ordered list $a_1, a_2, a_3, \dots$. An arithmetic sequence adds a fixed common difference $d$ each step.

$a_n = a_1 + (n-1)d$

For $2, 5, 8, 11, \dots$, the common difference is $d = 3$.

An arithmetic series is the sum of the first $n$ terms. The sum equals the number of terms times the average of the first and last:

$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}\big(2a_1 + (n-1)d\big)$

Young Gauss used this to add $1$ to $100$ instantly: $\frac{100}{2}(1 + 100) = 5050$.

Geometric Sequences & Comparison

A geometric sequence multiplies by a fixed common ratio $r$ each step:

$a_n = a_1 r^{n-1}$

For $3, 6, 12, 24, \dots$, the ratio is $r = 2$. Its finite sum is:

$S_n = a_1 \frac{r^n - 1}{r - 1} \quad (r \neq 1)$

When $|r| < 1$, the infinite series converges to $\frac{a_1}{1 - r}$ ($1 + \frac{1}{2} + \frac{1}{4} + \dots = 2$).

Common pitfall: Mixing up formulas. Always identify whether terms add or multiply before applying a sum formula.