Sequences and Series

$\{\ a,\ b,\ c,\ d\ ...\ \}$

• $n^{th}$ iteration: $t_n$
• $n$ can start at $0$ or $1$

Defining sequences

• sequence rule in terms of $n$ e.g. $t_n = 2n$

• recurrence relation - value of term is derived from previous term (recursion)
  e.g. $\ t_n=t_{n-1}+7,\ t_1=4$

  • $t_1=4$
  • $t_2=4+7=11$
  • $t_3=11+7=17$
  • $\dots$

Arithmetic sequences

$t_n = a+(n-1)d$

where
$a=$ first term
$d=$ common difference

Arithmetic mean

$m_a = \frac{a+b}{2}$

Arithmetic series

$\sum$ of all terms in an arithmetic sequence.

e.g. A sequence is defined by $t_n = a+(n-1)d$

$s_n = {\frac{n}{2}}(2a+(n-1)d)$ <–sum of $n$ terms, including $t_n$ (works only with $t_1$)
where

• $a=t_1$ (first term)
• $l=t_n$ (last term)

Geometric sequences

$t_n=ar^{n-1}$
where
$a =$ first term
$r =$ common ratio of successive terms ($r=\frac{t_k}{t_{k-1}}$)

Geometric mean

$m_g$ of $a$ and $b$ is $\sqrt{ab}$

If $a, c, b$ are positive and consecutive terms in a geometric sequence, then:
${\frac{c}{a}}={\frac{b}{c}}\therefore c=\sqrt{ab}$,

Geometric series

$\sum$ of all terms in a geometric sequence.

e.g. $s_n = a + ar + ar^2 + ar^2 + \dots + ar^{n-1}$

$rs_n = ar+ar^2+ar^3+ar^4+\dots+ar^{n}$

$rs_n - s_n = -a + ar^n$

$s_n = {\frac{a(r^n-1)}{r-1}}$
or
$s_n = {\frac{a(r^n-1)}{r-1}}$

Infinite series

If $-1 < r -1$, the infinite geometric series $a+ar+ar^2 \dots$ is convergent.

Sum to infinity is given by

$s_\infty={\frac{a}{1-r}}$

Tennis ball question - remember down and up strokes. Multiply down strokes by 2, subtract 1.