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$m_a = \frac{a+b}{2}$ 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$s_n = {\frac{n}{2}}(2a+(n-1)d)$ <--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 ($r=\frac{t_k}{t_{k-1}}$) 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${\frac{c}{a}}={\frac{b}{c}}\therefore c=\sqrt{ab}$, 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$s_n = {\frac{a(r^n-1)}{r-1}}$ 68or 69$s_n = {\frac{a(r^n-1)}{r-1}}$ 70 71## Infinite series 72 73If $-1 < r -1$, the infinite geometric series $a+ar+ar^2 \dots$ is convergent. 74 75Sum to infinity is given by 76 77$s_\infty={\frac{a}{1-r}}$ 78 79Tennis ball question - remember down **and** up strokes. Multiply down strokes by 2, subtract 1.