Section 14

Sequences, finite series, and recurrences

fundamentalviz

a_{n} = a_{1} + (n - 1) d

Each term differs from the previous one by a constant.

fundamental

S_{n} = \frac{n}{2} (a_{1} + a_{n})

Sum of the first n terms of an arithmetic sequence.

fundamentalviz

a_{n} = a_{1} r^{n - 1}

Each term is obtained by multiplying by a constant ratio.

fundamental

S_{n} = a_{1} \frac{1 - r ^{n}}{1 - r} (r \neq = 1)

Sum of n terms of a geometric sequence.

intermediateviz

k = 0 \sum \infty a r^{k} = \frac{a}{1 - r}, ∣ r ∣ < 1

Sum of a convergent geometric series.

intermediate

\sum (a_{k} + b_{k}) = \sum a_{k} + \sum b_{k}; \sum c a_{k} = c \sum a_{k}

Fundamental rules of sigma notation.

advancedviz

a_{n} = c_{1} a_{n - 1} + c_{2} a_{n - 2}; r^{2} - c_{1} r - c_{2} = 0

Uses a characteristic equation to solve homogeneous recurrences.