Warmup

1
Warmup

• P. 227 21, 22

2
Homework Check

• p. 212 5a,b
• p. 219 4
• p. 225 8

3

• Chapter 4
• Sequences and Series
• Arithmetic and Geometric Sequences
• 4-4
• Today we will
• Learn to identify has a common difference
• Write explicit and recursive formulas for
  arithmetic sequences
• Lear how to determine the number of terms in an
  arithmetic sequence

SEQUENCES
4

• 2/5 19/10 17/5 49/10
• When you subtract the previous term from current
  term you get a term called the common difference
• d an an-1
• d 19/10 2/5 19/10 4/10 15/10 3/2
• 19/10 3/2 19/10 15/ 10 34/10 17/5

5

• 2/5 19/10 17/5 49/10
• If you add the common difference to a term you
  should get the next term in the sequence
• d 3/2
• 19/10 3/2 19/10 15/ 10 34/10 17/5

6
Arithmetic Sequences

• 0 3 6 9 12
• 3 3 3 3
• 10 5 0 -5 -10
• -5 -5 -5 -5
• 3, -5 called Common Difference.
• Arithmetic Sequences have a common difference.

7
Example

• What is the common difference for the sequence
• 9/2 43/10 41/ 10 39/10 37/10
• d 43/10 9/2 43/10 45/10 -2/10 or -1/5
• What is the next term in the sequence?
• 37/20 2/10 35/10 or 7/2

8
Recursive Formula for Arithmetic Sequences

• The Recursive formula uses the previous term in
  the sequence.
• an an-1 d
• Example
• 3 7 11 15 19
• 4 4 4 4 d 4
• Recursive Formula
• an an-1 4

9
Example
Explicit formula for Arithmetic sequence
a1 (n-1)d
10

• Explicit Formula tells you how to find any value
  in a sequence.
• For any Arithmetic Sequence you can create an
  explicit formula using the first term and the
  common difference
• an a1 (n-1)d
• Where a1 is the first term,
• n is the term number and d is the common
  difference

11

• Example
• For a sequence with a1 3 and d 4
• an 3 (n-1)4
• Find the 10th term.
• a10 3 (9)4 39
• Find the explicit formula for
• 6 8 10 12 14
• d 2, a1 6 an 6 (n 1) 2
• a10 6 (9)2 6 18 24
• a200 6 (199)2 6 398 404

12
Example

• What is Explicit formula for
• 7/8 ¾ 5/8 ½
• This equals 7/8 6/8 5/8 4/8
• d -1/8
• an 7/8 (n-1)(-1/8)
• Can simplify to
• an 7/8 - 1/8(n) 1/8
• an 1 1/8n

13

• For Arithmetic Sequence
• 5 3 1 -1
• d -2
• Explicit formula an 5 (n-1)-2
• 5 -2n 2 7 2n
• You can create a recursive formula using the
  common difference
• an an-1 d
• Recursive formula an an-1 - 2

14
Find the Explicit and Recursive formula for

• 6 9 12 15 18
• Common Difference 3
• Explicit an 3 (n-1)(3)
• 3 3n -3
• 3n
• Recursive an an-1 3

15
How to determine the number of terms in an
arithmetic sequence

• 1, 5, 9, 13,., 461
• Find the common difference
• 9 5 4, 13-9 4, d 4
• 2) Use the explicit formula for arithmetic
  sequences
• an a1 (n-1)d
• an is the last term in the sequence
• an 461, a1 1, d 4

16
How to determine the number of terms in an
arithmetic sequences

• 1, 5, 9, 13,., 461
• Plug 461, 1 and 4 into the formula and solve for
  n
• 461 1 (n-1)4
• 461 1 4n -4
• 461 4n 3
• 3 3
• 464 4n
• 4 4
• 116 n
• There are 116 terms in this
• sequence

17
Find the number of terms in the following
sequence

• 45, 32, 19, 6, -137
• 1) d -13
• 2) a1 45, an -137
• -137 45 (n-1)(-13)
• -137 45 -13n 13
• -137 58 13n
• -195 -13n
• 15 n

18
Homework

• p. 233-23419-22
• Web Activity of the Week
• http//www.interactivestuff.org/sums4fun/sequences
  .html
• Go through 10 sequences. Write the the sequence
  and the rule. (3 points)