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Aim #83: How do we identify number sequences?
4-13-15
Do Now: Find the next term in the following number patterns.
-4, 8, -16, 32, ________
1, 5, 9, 13, ______
A sequence is a list of ordered elements.
Example: { 1, 2, 4, 8, 16, ...}
Each term in the sequence is indexed. { a1, a2, a3, a4, a5, ...}
So in this example a3 = 4.
Do you agree that this sequence can be identified as an = 2n ?
In general, a sequence is defined by a function f from a domain of natural
numbers to a range of real numbers that satisfy the equation f(n) = an.
Example: f(n) = 2n + 1
f: N
Z f(1) = a1
{3, 5, 7, 9, 11, ...}
Exercise: Write the sequence from the given explicit formula
f(n) = 3n - 4 for n ≥ 1.
Exercise: Find the next term in the sequence { 3, 6, 9, 12, ... }.
• Is it possible to find the 25th term in the sequence without writing them all
out?
Write a formula using either function notation or an notation to determine the
nth term of the sequence. Basically, write an equation that will find any nth
value.
Let's examine the two sequence formulas.
f(n) = 3n for { n Whole }
f(n) = 3n-1 for { n Natural }
and
Because there is mathematically no difference between the formulas, for
ease later on we will always use the domain { n Natural }. It keeps a1 = f(1).
Term #
1
2
3
4
5
6
7
Term
1
2
4
8
16
32
64
f(4) = a4 = 8
Example: Write the formula for the sequence above in function notation.
Example: Consider the equation that follows a "plus 3" pattern:
4, 7, 10, 13, 16, ...
a) Write the formula for the sequence in an form.
b) Graph the terms of the sequence as ordered pairs ( n, an ) on the
coordinate plane. What do you notice about all the points?
Exercise: Consider a sequence that follows a "minus 5" pattern:
30, 25, 20, 15, ...
a) Write the formula for the nth term of the sequence. Be sure to specify
what value of n your formula starts with.
b) Using the formula, find the 20th term.
Exercise: Consider the sequence that follows a "times 5" pattern:
1, 5, 25, 125, ...
a) Write a formula for the nth term of the sequence. Be sure to specify
what value of n your formula starts with.
b) Using the formula, find the 10th term.
f(9) = 59 =
c) Graph the terms of the sequence as ordered pairs ( n, f(n) ) on a
coordinate plane.
Exercise: Consider the sequence formed by the square numbers:
a) Write a formula for the nth term of the sequence. Be sure to specify what
value of n your formula starts with.
b) Using the formula, find the 50th term.
c) Graph the terms of the sequence as
ordered pairs ( n, f(n) ) on the
coordinate plane.
Exercise: A standard letter-sized piece of paper is 8.5 by 11 inches.
a. Find the area of one piece of paper
b. If the paper were folded completely in half, what would be the area of
the resulting rectangle?
c. Write the formula for a sequence to determine the area of the paper after n
folds.
d. What would the area be after 7 folds?
Let's sum it up!
A sequence can be thought of as an ordered list of elements. To
define the pattern of a sequence, an explicit formula is often given,
and unless specified otherwise, the first term is found by substituting
1 into the formula.
HW #83 Answers:
1. a. f(n) = 8n + 2 starts at n = 0
f(n) = 8n - 6 starts at n = 1
b. f(100) = 802
f(100) = 794
2. an = n3
3. 2, 8, 14, 20, 26
4. 1, 1/3, 1/9, 1/27, 1/81
5. -3, 3, -3, 3, -3
Mixed Review:
x = 22.5