Spec / sequences.mdon commit initial commit (c0222d9)
   1# Sequences and Series
   2
   3$\{\ a,\ b,\ c,\ d\ ...\ \}$
   4
   5
   6- $n^{th}$ iteration: $t_n$
   7- $n$ can start at $0$ or $1$
   8
   9## Defining sequences
  10- sequence rule in terms of $n$ e.g. $t_n = 2n$
  11
  12- recurrence relation - value of term is derived from previous term (recursion)
  13e.g. $\ t_n=t_{n-1}+7,\ t_1=4$
  14  - $t_1=4$
  15  - $t_2=4+7=11$
  16  - $t_3=11+7=17$
  17  - $\dots$
  18
  19## Arithmetic sequences
  20
  21$$ t_n = a+(n-1)d$$
  22
  23where
  24$a=$ first term
  25$d=$ common difference
  26
  27### Arithmetic mean
  28
  29$$m_a = {{a+b} \over 2}$$
  30
  31### Arithmetic series
  32
  33$\sum$ of all terms in an arithmetic sequence.
  34
  35e.g. A sequence is defined by $t_n = a+(n-1)d$
  36
  37$s_n = {n \over 2}(2a+(n-1)d)$ <--sum of $n$ terms, including $t_n$ (works only with $t_1$)
  38where
  39- $a=t_1$ (first term)
  40- $l=t_n$ (last term)
  41
  42## Geometric sequences
  43
  44$$t_n=ar^{n-1}$$
  45where
  46$a =$ first term
  47$r =$ common ratio of successive terms ($r={t_k \over t_{k-1}}$)
  48
  49### Geometric mean
  50
  51$m_g$ of $a$ and $b$ is $\sqrt{ab}$
  52
  53If $a, c, b$ are positive and consecutive terms in a geometric sequence, then:
  54${c \over a } = {b \over c} \therefore c = \sqrt{ab}$,
  55
  56### Geometric series
  57
  58$\sum$ of all terms in a geometric sequence.
  59
  60e.g.  $s_n = a + ar + ar^2 + ar^2 + \dots + ar^{n-1}$
  61
  62$rs_n = ar+ar^2+ar^3+ar^4+\dots+ar^{n}$
  63
  64$rs_n - s_n = -a + ar^n$
  65
  66---
  67$s_n = {{a(r^n-1)} \over {r-1}}$
  68or
  69$s_n = {{a(r^n-1)}\over r-1}$
  70
  71## Infinite series
  72
  73If $-1 < r -1$, the infiniteg eometric series $a+ar+ar^2 \dots$ is convergent.
  74
  75Sum to infinity is given by
  76
  77$$s_\infty={a \over {1-r}}$$
  78
  79Tennis ball question - remember down **and** up strokes. Multiply down strokes by 2, subtract 1.