AP Precalculus: Series (Arithmetic & Geometric) Formulas

1. Series, Sigma Notation & Types

Series: Sum of terms in a sequence \( a_1 + a_2 + a_3 + \cdots \)
Sigma: \( \sum_{k=m}^{n} a_k \) = \( a_m + a_{m+1} + \cdots + a_n \)
Arithmetic: Each term differs by \( d \)
Geometric: Each term is multiplied by \( r \)

2. Arithmetic Series (Partial Sum)

Explicit Sum: \( S_n = a_1 + a_2 + \cdots + a_n \)
Short Formula: \( S_n = \frac{n}{2}(a_1 + a_n) \)
Or: \( S_n = \frac{n}{2}[2a_1 + (n-1)d] \)
\( n \) = number of terms, \( d \) = common difference

3. Geometric Series (Partial Sum)

For \( r \neq 1 \): \( S_n = a_1 \frac{1 - r^n}{1 - r} \)
\( S_n \) = sum of first \( n \) terms, \( r \) = common ratio

4. Infinite Geometric Series

Converges if \( |r| < 1 \): \( S = \frac{a_1}{1 - r} \)
Diverges if \( |r| \geq 1 \)

5. Partial Sums & Mixed Series

Partial sum: sum of first \( n \) terms; use above \( S_n \) formulas
To identify type: check for constant difference (\( d \)) or ratio (\( r \))
If not clear, expand terms or analyze recursively

6. Repeating Decimal as Fraction

Write repeating decimal as infinite geometric series
Find \( a_1 \) (first repeating block), \( r \) (decimal shift), and use \( S = \frac{a_1}{1 - r} \)
Example: \( 0.\overline{27} = \frac{27}{99} \)