GOAL
BUILDING FUNCTIONS
Write arithmetic and geometric sequences both recursively and
with an explicit formula, use them to model situations, and translate
between the two forms.★
Tuesday Schedule -
· Return quizzes & discuss
· Review HW
· Notes
· HW
Precalculus 9.2 9/17/13
Notes on Arithmetic Sequences HW: Pg 674 #2 – 5, 25, 27, 29, 37, 39
I. Arithmetic Sequences
A. Definition: A sequence whose successive terms differ by the same nonzero number d, called the
common difference. (this would be the slope of a linear function)
B. Refer to the patterns worksheet and write a function to find the nth term of the sequence for #1 and 2.
a. Create a table
b. Find the common difference (slope)
c. Write an equation in point-slope form and solve
d. Write function using subscript notation
iii. Problem #1: an=3n-2 Problem #2: an=-6n+11
C. In other words, the formula to find the nth term of an arithmetic sequence is LINEAR.
II. Notation
A. Using point-slope form to write the formula for the nth term of the arithmetic sequence.
Let’s convert this into a point-slope form that uses subscript notation:
i. y-y1=mx-x1 → an-a1=d(n-1)
ii. The slope is the common difference d and the ordered pair x1,y1 is the first term of the
sequence.
iii. Now solve this equation for an to get an=a1+d(n-1)
B. You can also use the slope-intercept form to find the nth term of the arithmetic sequence.
i. y=mx+b → an=dn+c
ii. The slope is the common difference d. Substitute a term into the equation to find c.
II. Examples:
A. Finding terms given a1 and d: find a formula for an and the first five terms of each arithmetic
sequence.
1. The first term is 7 and the common difference is -3
2. a1=-12, d=5
B. Finding terms of an arithmetic sequence
1. Find a13 and an for the arithmetic sequence -3, 1, 5, 9, …
2. Find a18 and an for the arithmetic sequence having a2=9 and a3=15
3. Find a1 given that the arithmetic sequence has the terms a8=-16 and a16=-40
4. Find the ninth term of the arithmetic sequence whose first two terms are 2 and 9
Pg. 674
9.2 Arithmetic Sequences
HW: Pg 674 #2 – 5, 25, 27, 29, 37, 39
Determine if the following sequences are arithmetic. If it is, find the common difference:
2. 10, 8, 6, 4, 2,..... 4. 6. -12, -8, -4, 0, 4, ...
Arithmetic d = -2 Arithmetic d = -½ Arithmetic d = 4
8. ln 1, ln 2, ln 3, ln 4, ln 5,...... 10. 12, 22, 32, 42, 52, ......
1, 4, 9, 16, 25...
Not Arithmetic Not Arithmetic
Find a formula for an for the given arithmetic sequences.
25. 27. a1 = 5, a4 = 15 29. a3 = 94, a6 = 85
d = -(5/2) d = (10/3) d = -3
an = (5/2)n + (13/2) an = (10/3)n + (5/3) a3 = a1 + 2d
94 = a1 + 2(-3)
a1 = 100
an = -3n + 103
Write the first 5 terms of the arithmetic sequence.
37. a1 = 2, a12 = 46 39. a8 = 26, a12 = 42
d = 4 d = 4
2, 6, 10, 14, 18,... a8 = a1 + 7d
26 = a1 + 7(4)
a1 = -2
-2, 2, 6, 10, 14, ...