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The diagram to the right shows the first 3 rows of seating in an amphitheater. The rows are arranged in rings around a central stage.
How many seats are there in the
first row?
second row?
third row?
How many seats will there be in the 9th row??
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In the example above, we see that the number of seats in each row is
We call this a sequence, which simply means a list of numbers in a specific order. Most interesting sequences follow a pattern, for example:
Checkpoint
What number comes next in the sequence
Select the correct option
Discussion
Give your own examples of sequences. For each one you come up with, list the first 4 terms and clearly explain the pattern.
Try and come up with examples that have different "types" of patterns.
Here are four different sequences with their first four terms and explanations of the pattern behind each.
5,8,11,14: Each term is 3 more than the one before.
3,6,12,24: Each term is twice the one before.
3,5,9,17: Each term is double the previous, minus one.
1,1,2,3: The first two terms are 1, and each subsequent term is the sum of the two preceding terms.
Since sequences are ordered, they always have a 1st,2nd,3rd term etc. To more easily talk about sequences and their terms, we use the notation un to mean the nth term in a sequence. For example, in the sequence
the first term is u1=11, the second term is u2=8, and so on.
Checkpoint
In the sequence
what is u3?
Select the correct option
One important type of sequences are what we call arithmetic sequences. Here are some examples:
Discussion
What do the 3 sequences above have in common?
For each sequence, the change from one term to the next is always the same.
Sequence 1:
so each time we add 1.
Sequence 2:
so each time we add 5.
Sequence 3:
so each time we add –6.
In every case the “step” between successive terms is identical, so all three follow the same pattern of fixed increments.
A sequence is said to be arithmetic if the difference between consecutive terms is constant. In other words, we are always adding or subtracting the same amount to go from one term to the next.
The constant amount that is added to go from one term to the next is called the common difference, and is denoted d.
Checkpoint
Only one of the following sequences is arithmetic. What is its common difference, d?
Select the correct option
Identifying arithmetic sequences
A sequence is arithmetic if the difference between consecutive terms is constant, ie un+1−un=d, the common difference, for all n. For example
is arithmetic but
is not since 9−5=4=5−2=3.
Exercise
An arithmetic sequence has consecutive terms 2+k,k−1,8−k. Find the value of k.
Select the correct option
Discussion
Imagine a sequence with first term u1=5, and common difference d=3.
Can you find the 2nd, 3rd and 4th terms?
How can we find the 50th term?
We start with u1=5 and add the common difference 3 each time.
To get from the 1st term to the 50th term we add 3 a total of 49 times:
Answer:
u2=8,u3=11,u4=14,u50=152.
An arithmetic sequence starts with a first term u1, and each consecutive term follows by adding d.
Notice that for each term un above, we add (n−1) times d, because we start at u1 (i.e. n=1). In general then:
General term
The nth term in an arithmetic sequence is given by
where u1 is the first term and d is the common difference.
Checkpoint
An arithmetic sequence has first term u1=55 and common difference d=−2. Find the 21st term.
Select the correct option
Discussion
Explain why the formula for un has (n−1)d and not nd - why is the formula not un=u1+nd?
For an arithmetic sequence each term differs from the previous one by d. To see why there are n−1 increments of size d from u1 to un, look at small n:
Each time you go from one term to the next you add one more d. To reach the nth term you make n−1 such “steps,” not n. Hence
If you tried un=u1+nd then even for n=1 it would give u1+d, which is wrong.
Exercise
Given that the 5th term of an arithmetic sequence is 20 and the 10th term is 35.
Find the common difference d of this sequence.
Find the first term u1 of the sequence.
Select the correct option
When we know the first term and common difference d, we can write an expression for the nth term in terms of n:
Take, for example, u1=5 and d=−3:
This simplified expression for un can then easily be used to find any term:
We call this the closed form for an arithmetic sequence.
Exercise
An arithmetic sequence has u1=−3 and d=−5.
State the closed form for this sequence.
Hence find u9.
Select the correct option
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