1. Arithmetic Sequences and Series
12.2
Arithmetic Sequences and Series
Arithmetic Sequences ▪ Arithmetic Series

2. Arithmetic Sequences
An Arithmetic Sequence is one in which each successive term is derived by adding or subtracting the same number.
Where:
Notice how “d” is always added
To each consecutive term.
This number “d” is called the
common difference.
etc…

3. How do we find the common difference “d”?
To find the common difference, subtract successive terms:
For example: a2 – a1 or a4 – a3
Example: Find the common difference, d, for the arithmetic sequence 20, 13, 6, –1, –8, …
Notice that “d” is
always the same.
This indicates an
arithmetic sequence.

4. Example: Finding Terms Given a1 and d
Find the first five terms for each arithmetic sequence.
(a) The first term is –14, and the common difference is 6.
Starting with a1 = –14, add d = 6 to each term to get the next term.

5. What if we want to find the nth term?
Using substitution, we can also write each term as multiples of “d” being added to the first term.
etc…

6. What if we want to find the nth term?
Continuing in this manner, we quickly notice that each multiple of “d” being added to a1 is always one less than the term (n) that we’re finding.
This gives us the formula for finding the
nth term of an arithmetic sequence:

7. Example: Finding Terms of an Arithmetic Sequence
Find a16 and an for the arithmetic sequence 23, 20, 17, 14, … . so

8. The Sum Formulas for Arithmetic Sequences
Formulas for finding the sum of n terms of an arithmetic sequence:
Where: n = # of terms to sum. E.g. S15  sum of 15 terms
d = common difference
a1 = first term
an = nth term  E.g. a15 = 15th term in sequence

9. Example: Using the Sum Formulas
(a) Evaluate S21 for the arithmetic sequence 48, 44, 40, 36, … . so
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10. Example: Using the Sum Formulas (cont.)
(b) Use a formula for Sn to evaluate the sum of the first 200 positive integers.
The first 200 positive integers form the sequence 1, 2, 3, 4, …, 200. Thus, n = 200, a1 = 1, and a200 = 200. so
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

11. Example: Using the Sum Formulas
The sum of the first 15 terms of an arithmetic sequence is 345. If a15 = 65, find a1 and d.
Sum formula
S15 = 345, a15 = 65
Formula for nth term of an arithmetic sequence.
a15 = 65, a1 = –19
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

12. Geometric Sequences and Series
12.3
Geometric Sequences and Series
Geometric Sequences ▪ Infinite Geometric Series ▪ Annuities

13. Arithmetic Sequences
A Geometric Sequence is one in which each successive term is derived by multiplying or dividing the same number.
Where:
Notice how “r” is always multiplied
to each consecutive term.
This number “r” is called the
common ratio.
etc…

14. How do we find the common ratio “r”?
To find the common ratio, divide successive terms:
For example:
Example: Find the common ratio, r, for the geometric sequence 2, 4, 8, 16, 32, …
Notice that “r” is
always the same.
This indicates a
Geometric Sequence.

15. Geometric Sequences
In much the same way as an arithmetic sequence, any term in a geometric sequence will always contain a power of (n-1) common ratios for each consecutive term written as a multiple of a1
This gives us a formula for finding the nth term of a finite geometric sequence:

16. Example 2 Finding Terms of a Geometric Sequence
Find a5 and an for the geometric sequence 6400, 1600, 400, 100, … .
First find r:

17. Example 3 Finding Terms of a Geometric Sequence
Find r and a1 for the geometric sequence with second term –18 and fifth term 486.

18. Geometric Sequences
Formula for finding the sum of a finite geometric sequence:

19. Example 6 Finding the Sum of the First n Terms
This series is the sum of the first 8 terms of a geometric sequence having a1 = 4 ∙ 51 = 20 and r = 5.
Using the summation formula where r ≠ 1, we have

20. Geometric Sequences
Formula for finding the sum of an infinite geometric sequence:
(Only if: -1 < r < 1)

21. Example 7 Summing the Terms of an Infinite Geometric Series
and
Using the formula …

22. Example 8 Finding the Sum of the Terms of an Infinite Geometric Series
The series converges because –1 < r < 1, so

23. 12.6
The Binomial Theorem
A Binomial Expansion Pattern ▪ Pascal’s Triangle ▪ Binomial Coefficients ▪ The Binomial Theorem ▪ kth Term of a Binomial Expansion

24. Binomial Expansion Pattern

25. Binomial Expansion Pattern

26. Binomial Expansion Pattern
Row 0
Row 1
Row 2
Row 3
Row 4

27. Binomial Expansion Pattern

28. 11.4 Example 1 Evaluating Binomial Coefficients (page 1030)
Evaluate each binomial coefficient.

29. Working some examples Evaluate the following examples on Smartboard:
(x+y)4
(2x-1)3
(2x-1)15  Waaaay too tedious to use Pascal’s triangle!

30. What is the kth term of…?
Finding a certain term using Binomial expansion: What is the 4th term in (x+y)6?