Algebra 2 / Unit 7: Sequences and Series
7.1 Arithmetic Sequences
Notes
An arithmetic sequence adds the same common difference $d$ each time.
- Explicit formula: $a_n = a_1 + (n - 1)d$.
- Find $d$ by subtracting any term from the next one.
- Example: $5, 8, 11, 14, \dots$ has $d = 3$, so $a_n = 5 + 3(n - 1) = 3n + 2$ and $a_{10} = 32$.
- Given two terms: $\displaystyle d = \frac{a_m - a_k}{m - k}$.
Practice
Find the 20th term of $7, 11, 15, 19, \dots$
$a_{20} = 7 + 19(4) = 83$. Use $n - 1 = 19$ steps, not 20.
An arithmetic sequence has $a_1 = 50$ and $d = -6$. Which term equals 2?
$50 - 6(n - 1) = 2 \Rightarrow n - 1 = 8 \Rightarrow n = 9$.
An arithmetic sequence has $a_4 = 13$ and $a_9 = 33$. Find an explicit formula.
$\displaystyle d = \frac{33 - 13}{9 - 4} = 4$ and $a_1 = 13 - 3(4) = 1$, so $a_n = 1 + 4(n - 1)$.
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