Download Number Patterns and Sequences

Number Patterns and Sequences
Intro
Patterns are found throughout nature. Patterns are also used by mathematicians, engineers and
scientists to help generalize problems with large amounts of data. In this lesson, we are going to
learn how to find basic number patterns and determine general formulas for each pattern.
Before we begin looking at patterns, let us take a look at the terminology:
Each number in the sequence is called
a term.
2, 4, 6, 8, 10, …
The three little dots (named an “ellipsis”)
means the sequence continues forever.
The collection of numbers is called a
sequence.
The pattern or rule for a sequence tells us the next term in the sequence. If we are trying to find,
say, the next term in the sequence, we can use our pattern/rule to determine the value.
For example, in the sequence above we can spot our pattern pretty easily. It seems our sequence
is increasing by 2 each time; this is called our common difference. If we wanted to find the next
term in our sequence, we would simply add 2 (common difference) to our last term: 10 + 2 = 12;
therefore the next term in this sequence is 12.
Sometimes, the pattern of a sequence is not so obvious.
Suppose you were given the following sequence: 1, 4, 9, 16, 25, …
The pattern for this sequence is a little harder to see. When you can not find a simple pattern, or
common difference, you might need to ask yourself what type of numbers appear in the sequence.
The numbers in this particular sequence seem to be square numbers; notice the 3rd-term is 9 and
the 4th-termis 16 and so on.
So, we can conclude that our next term, the 6th-term, will be 36.
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Knowledge of number sense, concepts and operations
Apply number concepts including primes, factors and multiples to build number sequences.
1
Take a look at this sequence: 1, 3, 5, 7, 11, 13, …
At first glance, it seems the pattern is the odd numbers or that the common difference is 2.
However, the rule breaks when we go from the 4th term to the 5th term. This particular pattern is a
bit more obscure, but if we are looking at the types of numbers in the sequence, we can see they
also happen to be prime numbers. Therefore, the next number is this sequence is 17.
There are many different types of sequences, each of whom have general formulas for finding
terms in their sequences.
In this lesson you will learn how to find the general formula for one called the arithmetic
sequence.
Arithmetic Sequences
An arithmetic sequence is a sequence which has a constant difference between terms. We can
say the sequence is made by adding the same value each time.
Let’s look at the following sequence,
1, 2, 3, 4, 5, …
If we wanted to find the next number in the sequence, we would need to find the “rule” which
generates the following term. In this case, we can see each term increases by 1 (our constant
difference); therefore the next term in the sequence is 6.
+1 +1 +1 +1 +1
1, 2, 3, 4, 5, 6
Here’s a second example,
3, 8, 13, 18, 23, …
To find the next term in the sequence, we need to find the rule which generates the following
term. The terms increase by 5 (our constant difference) each time; therefore the next term in our
sequence is 28.
+5
…
3, 8, 13, 18, 23, 28
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Knowledge of number sense, concepts and operations
Apply number concepts including primes, factors and multiples to build number sequences.
2
Finding the nth-term in arithmetic sequences
Finding the next term in a sequence is easy when you find the rule. But suppose we wanted to
find a term other than the next, say, the 10th term. We still use our rule, but written in a slightly
different manner in order to generalize it.
The explicit formula for an arithmetic sequence is: a n = a1 + (n − 1)d
Now, I know the formula looks at bit scary, but it is actually quite easy to follow: n stands for the
term and d stands for the common difference.
Look at this example: 2, 4, 6, 8, 10, …
a1 is always the first term in the sequence: in our example this is 2
d is the common difference in our sequence: in our example, this is also 2 (note: the first term and
common difference will not always be the same number).
Substituting these values into our formula yields the following: an = 2 + (n – 1)2
an = 2 + (n – 1)2
an = 2 + 2n – 2
Using the distributive property on “(n-1)2”, I get “2n-2”
an = 2n
I subtracted “2 – 2” which left me with “2n”
Therefore, my general rule for this sequence is 2n. This means that no matter what term I am
asked for in the sequence, I can use the general rule, 2n, to find my result.
For example, if I am asked the value of the 100th term in the sequence I can simply substitute 100
for n to find my result: 2(100) = 200. Likewise, if I am asked to find the 88th term in the sequence:
2(88) = 176.
The general rule is often referred to as the “nth-term” of a sequence.
1
Knowledge of number sense, concepts and operations
Apply number concepts including primes, factors and multiples to build number sequences.
3
Let’s try some!
Example 1
Find the nth-term (general rule) for the given sequence: 3, 6, 9, 12, 15,…
Solution
Simply by looking at our sequence, we can determine our common difference, 3. In order to find
our nth-term, we can substitute our values into the formula: a n = a1 + (n − 1)d
an = 3 + (n – 1)3
Substituting our values from our sequence.
an = 3 + 3n – 3
Using the distributive property.
an = 3n
Simplifying.
Therefore, our nth-term for this sequence is 3n.
Example 2
Find the nth-term for the given sequence: 5, 8, 11, 14, 17, …
Solution
The common difference of this sequence is 3.
a n = a1 + (n − 1)d
an = 5 + (n – 1)3
Substituting our values from our sequence.
an = 5 + 3n – 3
Using the distributive property.
an = 3n + 2
Simplifying: “5 – 3 = 2”.
Therefore, the nth-term for this sequence is 3n + 2.
1
Knowledge of number sense, concepts and operations
Apply number concepts including primes, factors and multiples to build number sequences.
4
Example 3
Find the 10th term for the given sequence: 3, 9, 15, 21, 27, …
Solution
The common difference is 6.
a n = a1 + (n − 1)d
an = 3 + (n – 1)6
Substituting our values from our sequence.
an = 3 + 6n – 6
Using the distributive property.
an = 6n - 3
Simplifying: “3 – 6 = -3”.
Therefore, the nth-term for this sequence is 6n - 3. But, the question asks for the 10th term, so we
can substitute: 6(10) – 3 = 57. The 10th term of the sequence is 57.
Example 4
Find the 56th term for the given sequence: 9, 5, 1, -3, -7, …
Solution
Notice this sequence is a bit different because the values are decreasing. The common difference
here is -4 because we are subtracting (instead of adding) 4 each time.
a n = a1 + (n − 1)d
an = 9 + (n – 1)-4
Substituting our values from our sequence.
an = 9 + -4n + 4
Using the distributive property; be careful, you are distributing a -4 which makes -4(-1) = 4.
an = -4n + 13
Simplifying: “9 + 4 = 13”.
Therefore, the nth-term for this sequence is -4n + 13. But, the question asks for the 56th term, so
we can substitute: -4(56) + 13 = -211. The 56th term of the sequence is -211.
This particular example was a bit more challenging because you are using negative numbers. Just
remember to pay particular attention to how you multiply, add or subtract negative numbers.
1
Knowledge of number sense, concepts and operations
Apply number concepts including primes, factors and multiples to build number sequences.
5