MTH 180 Concepts

MTH 180 Concepts#

Sequence#: a function whose domain is the set of positive integers
ex: ${1, 2, 3, 4, \dots}$

The general, or $n^{t h}$ term, is denoted as $a_{n}$

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Write out the first five (5) terms of the sequence…

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Write the $n^{t h}$ term of the sequence ${a_{n}}$ suggested by the pattern

Recursive relation#: a sequence for which the given is the first term together with a rule for obtaining any $a_{n + 1}$ from the preceding term $a_{n}$ for $n \geq 1$

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Find the first five (5) terms of the recursively defined infinite sequence

c a n b e f i n i t e

1 + 3 + 5 + 7

c a n b e i n f i n i t e

1 + 3 + 5 + 7 + \dots

1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots

Summation / Sigma Notation

\begin{array}{r} k - i n d e x o f \\ s u m m a t i o n \end{array}

$n$ = stopping point
$\sum$ = $s i g m a$ or “sum of”
$k$ = starting point
$a_{k}$ = formula for each term

Example

\sum_{k = 1}^{n} a_{k} = a_{1} + a_{2} + a_{3} + a_{4} + \dots + a_{n - 1} + a_{n}

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Write out each sum (expand and simplify)

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Express each sum using summation notation

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Find the sum for each sequence

$2$	$- 4$	$6$	$- 8$	$10$	$\dots$	$n$
$a_{1}$	$a_{2}$	$a_{3}$	$a_{4}$	$a_{5}$	$\dots$	$a_{n}$