Sequences and Series

Sequences

A sequence is an ordered list of numbers: $a_1, a_2, a_3, \ldots$. Algebra 2 focuses on two types.

Arithmetic sequence

Each term differs from the previous by a constant $d$ (the common difference).

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

Example. $3, 7, 11, 15, \ldots$ has $a_1 = 3$ and $d = 4$. So $a_{10} = 3 + 9 \cdot 4 = 39$.

Geometric sequence

Each term is the previous times a constant $r$ (the common ratio).

$a_n = a_1 \cdot r^{\,n-1}$

Example. $2, 6, 18, 54, \ldots$ has $a_1 = 2$ and $r = 3$. So $a_8 = 2 \cdot 3^7 = 4374$.

Series — summing a sequence

Arithmetic series

$S_n = \dfrac{n}{2}\,\bigl(a_1 + a_n\bigr)$

Read it: the average of the first and last term, times the number of terms.

Example. Sum of $1 + 2 + 3 + \cdots + 100$. $n = 100$, $a_1 = 1$, $a_{100} = 100$:

$S_{100} = \dfrac{100}{2}(1 + 100) = 50 \cdot 101 = 5050$

Geometric series (finite)

$S_n = a_1 \cdot \dfrac{1 - r^n}{1 - r}, \quad r \neq 1$

Example. Sum of $2 + 6 + 18 + 54 + 162$. $a_1 = 2$, $r = 3$, $n = 5$:

$S_5 = 2 \cdot \dfrac{1 - 243}{1 - 3} = 2 \cdot \dfrac{-242}{-2} = 242$

Infinite geometric series

If $|r| < 1$, the infinite series converges:

$S_\infty = \dfrac{a_1}{1 - r}$

If $|r| \geq 1$, it doesn't converge.

Example. $1 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \cdots$, with $a_1 = 1$, $r = 1/2$:

$S_\infty = \dfrac{1}{1 - 1/2} = 2$

Recursive vs. explicit

Explicit: plug $n$ in directly. $a_n = 3 + 4(n-1)$.

Recursive: define each term from the previous. $a_1 = 3$, $a_n = a_{n-1} + 4$.

Both describe the same sequence; pick whichever is convenient.

Key takeaways

• Arithmetic: constant difference $d$; $a_n = a_1 + (n - 1)d$.
• Geometric: constant ratio $r$; $a_n = a_1 r^{n-1}$.
• Arithmetic sum: average of endpoints times count of terms.
• Geometric sum: $a_1 \dfrac{1 - r^n}{1 - r}$; the infinite version converges only when $|r| < 1$.