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Types of Number Patterns
There are three common number sequence patterns:
1. Arithmetic Sequences
2. Quadratic Sequences
3. Geometric Sequences
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GEOMETRIC SEQUENCE
❖A Geometric Sequence is a sequence of numbers where there is a constant
common ratio(r).
❖In the geometric sequence we can determine the constant ratio (r) from:
𝑇2
𝑇3
𝑟=
=
𝑇1
𝑇2
❖To get the next term in the sequence we use the formula 𝑇𝑛𝑒𝑥𝑡 = 𝑇𝑝 × 𝑟
ACTIVITY 1
Determine the constant ratios for the following geometric sequences and write down
the next three terms in each sequence:
1. 5; 10; 20; . . .
1
1 1
2. ; ; ; . . .
2
4 8
3. 3𝑝 ; 3𝑝2 ; 9𝑝3 . . .
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The general geometric sequence can be expressed as:
𝑇𝑛 = 𝑎𝑟 𝑛−1
ACTIVITY 2
Determine the general formula for the nth term of each of the following geometric
sequences:
1. 5; 10; 20; . . .
1
1 1
2. ; ; ; . . .
2
4 8
3. 𝑝 ; 3𝑝2 ; 9𝑝3 . . .
Given a geometric sequence with second term
1
and ninth term 64.
2
a) Determine the value of r.
b) Find the value of a.
c) Determine the general formula of the sequence.
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SUM OF A GEOMETRIC SEQUENCE
❖Geometric Series is the sum of the terms of a Geometric Sequence.
❖The sum of a Geometric sequence can be calculated using the formula:
𝑆𝑛 =
𝑎(𝑟 𝑛 −1)
𝑟−1
or
𝑆𝑛 =
𝑎(1−𝑟 𝑛 )
1−𝑟
where 𝑟 ≠ 𝑟
ACTIVITY 3
Given the geometric sequence 1; −3; 9; . . . determine:
a) The eighth term of the sequence.
b) The sum of the first eight terms of the sequence.
ACTIVITY 4
The eighth term of a geometric sequence is 640. The third term is 20. Find the sum of
the first 7 terms.
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ACTIVITY 5
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THE SUM TO INFINITY OF A GEOMETRIC SEQUENCE
❖The sum of an infinite geometric series only exists for convergent sequences.
❖A geometric series will converge (the sum will approach a specific value), if the constant
ratio is a number between -1 and 1.
❖If (r) is greater than 1 or less than -1, the sum of the infinite geometric series cannot be
evaluated.
❖We can calculate the sum to infinity of a convergent geometric series by using the
following formula:
ACTIVITY 6
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ACTIVITY 7
P.Q
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SIGMA NOTATION
❖Writing out the terms of a series can be tedious and time consuming.
❖In this section we will introduce a new notation, called sigma notation, which is
quicker and easier to write down
❖We use the Greek letter Σ (sigma) to indicate the sum of a series.
❖A series written in a sigma notation takes the following form:
Example:
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