Download Trig-12-Sequences-sigma notation

10/1/2013
Definition:
Sequences & Sigma Notation
Trigonometry
Radnor High School
A sequence is function whose domain is the set of
natural (counting) numbers.
The outputs of a sequence are called terms.
For convenience,
convenience we often refer to the terms of the
sequence as the sequence involved.
Normally, sequence is written as a list of numbers
written in a definite order: a1, a2, a3,…, an,…
The number a1 is called the first term, a2 is called the
second term and so on.
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A sequence can be represented as a list of numbers, or
defined by a closed form rule for the nth term, or
defined recursively (first term(s) are given and rule to
find the next term).
Example 1:
Write the next three terms of the following sequences:
a ) 4, 8, 14, 22, 32, 44, ...
For example:
a) 1,
1 5,
5 7,
7 3,
3 4,
4 8,
8 10,
10 ….
b) 3, 7, 11, 15, 19, …
b) 3, 6, 12, 24, 48, ....
c) an = 3n +1
d) b1 = 5, bn+1 = 3bn + 4
Note:
Certain sequences have patterns between consecutive
terms. Knowing these patterns, we can write a rule for the
general nth term of the sequence.
patterns frequently
q
y occur are:
Some of the p
c)
1 4 7 10
,  , ,  ,...
3 9 27 81
Example 2:
Write a closed form rule for the nth term of the
sequence below:
2, 5, 12, 19, 26, 33, ...
a) The first differences are constant (add/subtract the
same number to get the next term)  linear function
b) The second differences are constant  quadratic
function (use regression on the calculator to find the rule)
c) Multiply the same number to get the next term.
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Example 3:
Example 4:
Write a closed form rule for the nth term of the
sequence below:
6,9,16, 27, 42, 61,...
Example 5:
Write a closed form rule for the nth term of the
sequence below:
3 5 7 9 11
, , , , ,...
5 9 13 17 21
Sigma Notation
Write the first five terms of each sequence below:
k: ending number for n
a) an  n 2
k
b) bn   1
n
n
n3
: add the terms
n i
c) cn   n  1 !
Example 6:
Evaluate each sigma notation by writing out the terms:
Evaluate each sigma notation by writing out the terms:
a)
k  
b)
j 1
6
k 2
 j
5
  n  2 
3
an: terms to add
Example 7:
n 3
b)
n
i: starting number for n
7
a)
a 
2
 3 
4
  3n   2 
n 0
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