SEQUENCES
A sequence is a
list of numbers that follows a pattern.
In other words, a sequence is a list of numbers that can be thought of
as being numbered. For example, the
sequence
3, 6, 9, 12, 15,
is a list of
multiples of 3. The first term is 3, the
second term is 6, etc.
Example:
What is the 20th term in the sequence?
Answer: 60
Solution: Look for a pattern.
|
Number of Term |
Term |
|
1 |
3 = 3 x 1 |
|
2 |
6 = 3 x 2 |
|
3 |
9 = 3 x 3 |
|
|
|
|
n |
n x 3 or 3n |
The 20th term is
therefore 20 x 3 = 60.
While this example
may seem easy, we want to practice using patterns to find solutions to many
problems. We can look at two special
kinds of sequences: arithmetic
sequences and geometric sequences. We
provide formulas, but suggest that if you look for patterns you often won't
need to memorize too many formulas.
Besides, many problems don't have a familiar formula that will work to
find the answer.
Arithmetic Sequence: A sequence in which each successive term is
obtained from the previous term by the addition of a constant. For example, in the sequence
2, 12, 22, 32,
the first term
is 2 and each additional term is found by adding 10 to the previous term. In general, the nth term is found by the
formula
N = a + (n - 1)d
where N is the value of the nth term, a is the first term, and d is the difference between any two
terms. However, by looking for a
pattern, you can usually find a term in a sequence without memorizing a
formula.
Example:
What is the 20th term in the arithmetic sequence:
5, 8, 11, 14
?
Answer: 62
Solution: Look for a pattern.
|
Number of Term |
Term |
|
1 |
5 |
|
2 |
8 = 5 + 3 |
|
3 |
11 = 5 + 3 + 3
= 5 + 2(3) |
|
4 |
14 = 5 + 3 + 3
+ 3 = 5 + 3(3) |
|
|
|
|
nth |
N = 5 + (n-1)3 |
Note that in
each case the number of 3s is one less than the number of terms in the sequence. Therefore, the 20th term is the first term,
5, with 3 added 19 times:
5 + (19)3 = 62
Using a
formula: The nth term in an arithmetic
sequence is a+(n-1)d where a is the first term and d is the difference. In this sequence 5 is the first term and the
difference between each two consecutive terms is 3.
N = a + (n - 1)d
= 5 + (20 - 1)3 = 62
Geometric Sequences:
A sequence in which each
successive term is obtained from the previous term by the multiplication of a
constant. For example, in the sequence
3, 6, 12, 24, 48,
the first term
is 3 and each additional term is found by multiplying the previous term by
2. In general, the nth term is found by
the formula
where N is the value of the nth term, a is the first term, and r is the ratio between successive
terms. However, again, by looking for a
pattern, you can usually find a term in a sequence without memorizing a
formula.
Example:
Find an expression for the 20th term in the geometric sequence:
5, 20, 80, 320
?
A. 5 + 4(20)
B. 5 + 
C. 5 x 
D. 5 x 
E. 5 x 
Answer: D
Solution: Look for a pattern.
|
Number of Term |
Term |
|
1 |
5 |
|
2 |
20 = 5 x 4 |
|
3 |
80 = 5 x 4 x 4
= 5 x 42 |
|
4 |
320 = 5x 4 x 4
x 4 = 5 x 43 |
|
|
|
|
nth |
N = 5 x 4 |
Note that in
each case the exponent is one less than the number of terms in the
sequence. Therefore the 20th term is
5 x 
= 5 x 
.
Of course, we
can also use the formula to find the 20th term:
N = 
= 5 x 
= 5 x 