Patterns and Sequences
I. Arithmetic Sequence
An arithmetic sequence is a sequence where every term after the first is obtained by adding a constant called
the common difference.
The sequences 1, 4, 7, 10, … and 15, 11, 7, 3, … are examples of arithmetic sequences since each one has a
common difference of 3 and –4.
Arithmetic Rule
•
•
•
•
𝑎𝑛=𝑎1+(𝑛−1)𝑑
𝑎1 is the first term in the sequence
n is the number of the term you are trying to determine
d is the common difference
𝑎𝑛 is the value of the term that are looking for
Example:
1. Use the arithmetic formula to determine the 100th term of the following sequence: 75, 25, -25, -75, -125,
…
𝑎1 = 75; 𝑛 = 100; 𝑑 = −50
𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑
= 75 + (100 − 1)(−50)
= −4875
2. Use the arithmetic formula to determine the 10th term of the following sequence: 5, 12, 19, 26, …
𝑎1 = 5; 𝑛 = 10; 𝑑 = 7
𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑
= 5 + (10 − 1)(7)
= 68
Arithmetic Means
The terms between any two consecutive terms of an arithmetic sequence.
Example:
1. Insert 4 arithmetic means between 5 and 25.
Since we are required to insert 4 terms, then there will be 6 terms in all.
Let 𝑎1 = 5 𝑎𝑛𝑑 𝑎6 = 25
5, 𝑎2 , 𝑎3 , 𝑎4 , 𝑎5 , 25
𝑎6 = 𝑎1 + (6 − 1)𝑑
25 = 5 + (5)𝑑
20 − 5 = 5𝑑
4=𝑑
𝑎2=9; 𝑎3=13; 𝑎4=17; 𝑎5=21
2. Insert 2 arithmetic means between 6 and 12.
Since we are required to insert 2 terms, then there will be 4 terms in all.
Let 𝑎1=6 𝑎𝑛𝑑 𝑎4 = 12
6, 𝑎2 , 𝑎3 , 12
𝑎4 = 𝑎1 + (4 − 1)𝑑
12 = 6 + (3)𝑑
12 − 6 = 3𝑑
2=𝑑
𝑎2 = 8; 𝑎3 = 10
II. Geometric Sequence
Geometric Sequence is a sequence in which each term is obtained by multiplying the preceding term by a fixed
number called the common ratio of the sequence.
Formula for the nth term
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•
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𝑎𝑛 =𝑎1 𝑟 𝑛−1
𝑎1 = for the first term
𝑎𝑛 = for the nth term
r = for the common ratio
n = for the number of terms from 𝑎1 𝑡𝑜 𝑎𝑛
Example:
1. Find the 7th and 10th terms of the following geometric sequence.
9, 3, 1, 1/3, 1/9, …
𝑎1 = 9 𝑎𝑛𝑑 𝑟 =
𝑎𝑛 = 𝑎1 𝑟 𝑛−1
1
3
1 7−1
1 6
1
𝑎7 = 9 ( )
= 9( ) =
3
3
81
1 10−1
1 9
1
𝑎10 = 9 ( )
= 9( ) =
3
3
2187
2. Find the 12th term of the following geometric sequence.
10, 30, 90, 270, 810, …
𝑎1 = 10 𝑎𝑛𝑑 𝑟 = 3
𝑎𝑛 = 𝑎1 𝑟 𝑛−1
𝑎12 = 10(3)12−1 = 10(3)11 = 1, 771, 470
III.
Fibonacci Sequence
The Fibonacci sequence is a series of numbers where a number is found by adding up the two numbers before
it. Starting with 0 and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth.
Written as a rule, the expression is:
𝑋𝑛 = 𝑋𝑛−1 + 𝑋𝑛−2
When we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the
Golden Ratio "φ" which is approximately 1.618034...
IV.
Harmonic Sequence
Harmonic sequence is a sequence of numbers a1, a2, a3,… such that their reciprocals 1/a1, 1/a2, 1/a3… form an
arithmetic sequence (numbers separated by a common difference).
The sequence 1/1, ½, 1/3, ¼, … 1/n, … is an example of harmonic sequence. Notice that the reciprocals of the
terms form the arithmetic sequence 2, 3, 4, 5, …, n, …
Example:
1. Find the 10th term of the harmonic sequence ½, ¼, 1/6, 1/8, …
• Get the reciprocal: 2, 4, 6, 8, …
• Use the formula 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑
• Substitute:
Therefore, 1/20 is the 10th term of the harmonic sequence ½, ¼, 1/6, 1/8