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Arithmetic and Geometric Sequences
Discrete function – a function that is defined for a set of numbers that can be listed.
We will begin by looking at sequences.
Sequences are ordered lists of numbers.
Sequences can be finite in size or infinite is size.
A term is a number in a sequence. Subscripts (index/indices) are used to identify the position of the terms in the
sequence.
In this course our sequences will begin with index ‘1’. However, sequences are often defined to start with an index of
zero.
The general term, 𝒕𝒏, is a formula that expresses each term as a function of its position in the sequence.
What would the general term be for the sequences below?
1, 2, 3, 4, 5...
2, 4,6,8...
1 1 1 1
1, , , , …
2 3 4 5
1, 1, 2, 3, 5, 8, 13, 21...
An arithmetic sequence is a sequence that has a common difference between any pair of consecutive terms.
Find a formula for the general term in each arithmetic sequence below:
a) 1, 4, 7, 10...
In general....
b) 5, 3, 1, -1, -3...
Example: What is the 33rd term in the sequence 18, 11, 4, -3...?
The following sequences are clearly not arithmetic.
1, 2,4,8,16,32 …
1,3,9,27 …
1 1 1
4,2, , , …
2 4 8
These are called geometric sequences. Geometric ratios have a constant ratio (as opposed to common difference)
between consecutive terms. We call the constant ratio r.
What would the general term look like in a geometric sequence?
Examples
1. Determine whether each sequence below is geometric. If it is find the general term.
a)
1
, 1,2,4,8 …
2
b) 2, −6,18, −54 …
8 16
…
9
c) 9,6,4, 3 ,
2. Determine the general term in a geometric sequence if the 5th term is 144 and the 10th term is 4608.