Monday, April 15th
Agenda:
• Continue with Ch.10­ Sequences and Series
• 10.2 & 10.3 Notes
• Practice questions (sheet)
• Hand back and go over midterm
10.2 Arithmetic Sequences
Temperature (oC)
Speed of Sound
1
2
3
4
332.1
332.7
333.3
333.9
What can you say about the speed of sound as the temperature increases by 1?
A sequence with a common difference, d, between consecutive terms is called an arithmetic sequence.
Other important components of an arithmetic sequences:
tn is the general term or the nth term of the sequence
a (t1) is the first term of the sequence n is the term number
d is the common difference between successive terms in an arithmetic sequence
Temperature (oC)
Speed of Sound
1
2
3
4
332.1
332.7
333.3
333.9
a
a+d
a + 2d
t3
a + 3d
t4
t1
t2
Other important components of an arithmetic sequences:
tn is the general term or the nth term of the sequence
a (t1) is the first term of the sequence n is the term number
d is the common difference between successive terms in an arithmetic sequence
Using these generalizations, can you come up with a
general formula for all arithmetic sequences?
The general term, tn, for the arithmetic sequence with the first term, a, and a common difference, d, is written as
tn = a + (n­1)d
ex. 1) For the sequence ­1, 2, 5, 8,..., find:
The general term
•
The 10th term
•
The 100th term
•
ex. 2) In the arithmetic sequence 0, 4, 8, ..., 44, which term is 44?
tn = a + (n­1)d
ex. 3) For an arithmetic sequence, t7 = 9 and t12 = 19. What is the first term and the general term of the sequence?
tn = a + d(n­1)
Practice/Homework:
10.1 & 10.2 on worksheet
10.1 : • 1 a,b,c,e,f,i • 2 (all)
• 3 a,b
• 4 (all)
10.2 : • 1 (all)
• 2 (all)
• 3 (all)
Tuesday, April 16th
Agenda:
• Discuss: Math Peer tutoring club
• Discuss: Thursday
• Ch. 10.3 Problems involving arithmetic sequences • Practice Questions on Arithmetic Sequences
• Begin Ch. 10.4 & 10.5 Geometric Sequences + Problems involving geometric sequences
10.3 Solving problems involving arithmetic sequences
ex.1) For a compact car, the cost of maintenance and repairs increased by $85 each year. If in the first year the amount was $120, how much was the maintenance at the end of year five? After year ten? 120, 205, 290, ...
Practice Questions (15 minutes):
Worksheet Section 10.3
#1­7 (homework)
Try #8 when finished.
10.4 Geometric Sequences
Ex. If a radioactive substance has a half­life of 12h, then at the end of 12h, only half of the radioactive material remains. If you begin with 100g, then the amount of radioactive material is shown in the chart:
Time (h)
Amount of material (g)
0
12
24
36
48
100
50
25
12.5
6.25
How can you describe the pattern for the sequence in the chart?
For the sequence 100, 50, 25, 12.5, 6.25,..., the ratio of consecutive terms is constant.
25 = 1
i.e. 50 = 1
2
50
100 2
This constant ratio is called the common ratio, r. In general, r = tn tn­1 When a sequence has a common ratio between consecutive terms it is called a geometric sequence.
A geometric sequence with first term, a, and common ratio, r, can be written in general as a, ar, ar2, ar3, ...
t1, t2, t3, t4, ...
The general term for the geometric sequence is given by
tn = a r n­1
Ex. 1 A geometric sequence is given by 5, 10, 20, 40,...
a) Find the nth term.
b) Find the seventh term.
c) What term in the sequence is 5120?
Ex. 2
The fifth term of a geometric sequence is 1050 and the seventh term is 26 250. Find the first term.