1. Objectives Recognize and extend an arithmetic sequence.
Find a given term of an arithmetic sequence.

3. Time (s) 1 2 3 4 5 6 7 8
Time (s)
Distance (mi)
Distance (mi)
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
+0.2
In the distance sequence, each distance is 0.2 mi greater than the previous distance. When the terms of a sequence differ by the same nonzero number d, the sequence is an arithmetic sequence and d is the common difference. The distances in the table form an arithmetic sequence with d = 0.2.

4. Example 1A: Identifying Arithmetic Sequences
Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms.
9, 13, 17, 21,…
Step 1 Find the difference between successive terms.
9, 13, 17, 21,…
You add 4 to each term to find the next term. The common difference is 4. +4

5. Example 1A Continued
Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms.
9, 13, 17, 21,…
Step 2 Use the common difference to find the next 3 terms.
9, 13, 17, 21,
25, 29, 33,…
an = an-1 + d +4
The sequence appears to be an arithmetic sequence with a common difference of 4. The next three terms are 25, 29, 33.

6. Example 1B: Identifying Arithmetic Sequences
Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms.
10, 8, 5, 1,…
Find the difference between successive terms.
10, 8, 5, 1,…
The difference between successive terms is not the same. –2 –3 –4
This sequence is not an arithmetic sequence.

7. The pattern in the table shows that to find the nth term, add the first term to the product of (n – 1) and the common difference.

8. Example 2A: Finding the nth Term of an Arithmetic Sequence
Find the indicated term of the arithmetic sequence.
16th term: 4, 8, 12, 16, …
Step 1 Find the common difference.
4, 8, 12, 16,…
The common difference is 4.
Step 2 Write a rule to find the 16th term.
an = a1 + (n – 1)d
Write a rule to find the nth term.
a16 = 4 + (16 – 1)(4)
Substitute 4 for a1,16 for n, and 4 for d.
= 4 + (15)(4)
Simplify the expression in parentheses. =
Multiply.
The 16th term is 64.
= 64
Add.

9. Introduction to Geometric Sequences and Series

10. Investigation: Find the next 3 terms of each sequence:
{3, 6, 12, 24, …}
{32, 16, 8, 4, …}

11. Geometric Sequences
Sequences that increase or decrease by multiplying the previous term by a fixed number
This fixed number is called r or the common ratio

12. Finding the Common Ratio
Divide any term by its previous term
Find r, the common ratio:
{3, 9, 27, 81, …}

13. Your Turn: Find r, the common ratio: {0.0625, 0.25, 1, 4, …}
{-252, 126, -63, 31.5, …}

14. Arithmetic vs. Geometric Sequences
Arithmetic Sequences
Increases by the common difference d
Addition or Subtraction
d = un – un–1
Geometric Sequences
Increases by the common ratio r
Multiplication or Division

15. 9.4 Compare Linear, Exponential, and Quadratic Models
Students will Compare Linear, Exponential, and Quadratic Models

16. Identifying from an equation:
Linear
Has an x with no exponent.
y = 5
y = 5x y = ½x
2x + 3y = 6
Exponential
Has an x as the exponent.
y = 3x + 1 y = 52x
4x + y = 13
Quadratic
Has an x2 in the equation; the
highest power is 2.
y = 2x2 + 3x – 5 y = x2 + 9
x2 + 4y = 7

17. Examples: LINEAR, QUADRATIC or EXPONENTIAL? a)y = 6x + 3
b)y = 7x2 +5x – 2
c)9x + 3 = y
d)42x = 8
Exponential Growth
Positive Quadratic Increasing Linear Exponential Growth

18. Identifying from a graph:
Linear
Makes a straight line
Exponential
Rises or falls quickly in one direction
Quadratic
Makes a U or ∩ (parabola)

19. Is the table linear, quadratic or exponential
Is the table linear, quadratic or exponential? All x values must have a common difference
Linear
Never see the same y value twice.
1st difference is the same for the y values
Exponential
y changes more quickly than x.
Never see the same y value twice.
Common ratio for the y values
Quadratic
See same y more than once.
2nd difference is the same for the y values

20. EXAMPLE 2
Identify functions using differences or ratios b. x
– 2
– 1 1 2 y
– 2 1 4 7 10
Differences: 3 3 3 3
ANSWER
The table of values represents a linear function.

21. EXAMPLE 2
Identify functions using differences or ratios
Use differences or ratios to tell whether the table of values represents a linear function, an exponential function, or a quadratic function.
a. x –2 –
y –6 –6 –4 0 6
First differences:
Second differences: 2 2 2
ANSWER
The table of values represents a quadratic function.

22. Step 2 Write the function.
There is a constant ratio of 7. The data appear to be exponential.
y = abx
Write the general form of an exponential
function.
y = a(7)x
Plug in the common ratio for b.
Plug in your initial (starting) amount for a.
This is your model.
y = 8(7)x

23. Step 3 Predict the e-mails after 1 week.
y = 8(7)x
Write the function.
= 8(7)7
Substitute 7 for x (1 week = 7 days).
Use a calculator.
= 6,588,344
There will be 6,588,344 s after one week.