Precalculus / Unit 10: Sequences, Series and Intro to Limits
10.1 Arithmetic Sequences and Series
Notes
An arithmetic sequence adds the same common difference $d$ each time.
- $a_n = a_1 + (n - 1)d$.
- The sum of the first $n$ terms: $\displaystyle S_n = \frac{n(a_1 + a_n)}{2}$.
- Example: $5, 9, 13, \dots$ has $d = 4$, $a_{20} = 5 + 19 \cdot 4 = 81$ and $\displaystyle S_{20} = \frac{20(5 + 81)}{2} = 860$.
Practice
What is the 15th term of $3, 7, 11, \dots$?
$a_{15} = 3 + 14 \cdot 4 = 59$.
An arithmetic sequence has $a_4 = 11$ and $a_9 = 26$. What is $a_1$?
Five steps add 15, so $d = 3$. Then $a_1 = 11 - 3 \cdot 3 = 2$.
What is the sum $5 + 9 + 13 + \dots + 81$?
There are $\displaystyle \frac{81 - 5}{4} + 1 = 20$ terms, so $\displaystyle S = \frac{20(5 + 81)}{2} = 860$.
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