Algebra 2 / Unit 7: Sequences and Series
7.3 Recursive Formulas and Sigma Notation
Notes
- A recursive formula gives the first term and a rule using the previous term: arithmetic $a_n = a_{n - 1} + d$, geometric $a_n = r \cdot a_{n - 1}$.
- Sigma notation writes a sum compactly: $\displaystyle \sum_{k=1}^{4} k^2 = 1 + 4 + 9 + 16 = 30$.
- The number under $\Sigma$ is where $k$ starts; the number on top is where it stops.
- Number of terms $=$ top $-$ bottom $+ 1$.
Practice
Evaluate $\displaystyle \sum_{k=1}^{4} (2k + 1)$.
$3 + 5 + 7 + 9 = 24$.
A sequence has $a_1 = 2$ and $a_n = 3a_{n - 1} - 1$. Find $a_4$.
$a_2 = 5$, $a_3 = 14$, $a_4 = 3(14) - 1 = 41$.
Which sigma expression equals $2 + 4 + 6 + \dots + 20$?
$2k$ runs from $2(1) = 2$ to $2(10) = 20$.
Ready for more? Take the Unit 7 test.