Transcript Document
9.2 – Arithmetic Sequences and Series
An introduction………… 1, 4, 7, 10, 13 9, 1, 6.2, 6.6, 7, 7.4
3, 6 35 12 27.2
3 9
Arithmetic Sequences ADD To get next term
Arithmetic Series Sum of Terms 2, 4, 8, 16, 32 9, 1, 1/ 4, 1/16, 1/ 64 62 20 / 3 85 / 64 9.75
Geometric Sequences MULTIPLY To get next term
Geometric Series Sum of Terms
Find the next four terms of –9, -2, 5, … Arithmetic Sequence 9 5 2 7 7 is referred to as the common difference (d) Common Difference (d) – what we ADD to get next term Next four terms…… 12, 19, 26, 33
Find the next four terms of 0, 7, 14, … Arithmetic Sequence, d = 7 21, 28, 35, 42 Find the next four terms of x, 2x, 3x, … Arithmetic Sequence, d = x 4x, 5x, 6x, 7x Find the next four terms of 5k, -k, 7k, … Arithmetic Sequence, d = -6k -13k, -19k, -25k, -32k
Vocabulary of Sequences (Universal) a 1 First term a n nth term n number of terms S n sum of n terms d common difference nth term of arithmetic sequence a n a 1
sum of n terms of arithmetic sequen ce S n n 2
a 1 a n
Given an arithmetic sequence with a 15 38 and d 3, find a .
x 38 15 NA -3 a 1 First term a n nth term n number of terms S n sum of n terms d common difference a n 38 a 1
1 5
1
X = 80
Find S 63 of 353 -19 ??
63 x 6 a 1 First term a n nth term n number of terms S n sum of n terms d common difference a n a 1
??
19
6 3 1
??
353 S n S 63 n 2
a 1 a n
63
19 2 S 63 1 052 1
Try this one: Find a 16 if a 1 1.5 and d 0.5
1.5
x 16 NA 0.5
a 1 First term a n nth term n number of terms S n sum of n terms d common difference a n a 16 a 1
1.5
16 1
0.
5 a 16 9
Find n if a n 633, a 1 9, and d 24 9 633 x NA 24 a 1 First term a n nth term n number of terms S n sum of n terms d common difference a n a 1
633
x 1
2 4 633 2 4x 24 X = 27
Find d if a 1 6 and a 29 20 x -6 20 29 NA a 1 First term a n nth term n number of terms S n sum of n terms d common difference a n a 1
20 6
29 1
x 26 28x x 13 14
Find two arithmetic means between –4 and 5 -4 5 4 NA x -4, ____, ____, 5 a 1 First term a n nth term n number of terms S n sum of n terms d a n 5 x common difference a 1 4
4 1
3 The two arithmetic means are –1 and 2, since –4, -1, 2 , 5 forms an arithmetic sequence
Find three arithmetic means between 1 and 4 1 4 5 NA x 1, ____, ____, ____, 4 a 1 First term a n nth term n number of terms S n sum of n terms d a n 4 common difference a 1
5
1
x 3 4 The three arithmetic means are 7/4, 10/4, and 13/4 since 1, 7/4, 10/4, 13/4 , 4 forms an arithmetic sequence
Find n for the series in which a 1 5, d 3, S n 440 5 y x 440 3 a 1 First term a n nth term n number of terms S n sum of n terms d common difference a n y a 1
x
1
3 440 440 880 x 2 5 3 x
x 1
3 2 3x
0 3x 2 7x 880 Graph on positive window X = 16 S n n 2
a 1 a n
4 40 x 2
5 y
The
sum
of the first
n
terms of an infinite sequence is called the
n
th partial sum.
S n
n
2 (
a
1
a n
)
Example 6. Find the 150 th 16, 27, 38, 49, … partial sum of the arithmetic sequence, 5,
a
1
5
d
11
c
6
a n
11
n
6
a
150
S
150
150
2
123, 675
Example 7. An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on. How many seats are there in all 20 rows?
d
1
c
a n
1
1
d
a
20
39
S
20
20
2
590
Example 8. A small business sells $10,000 worth of sports memorabilia during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 19 years. Assuming that the goal is met, find the total sales during the first 20 years this business is in operation.
a
1
10,000
d
7500
c
2500
a n
1
S
20
20
2
1
d
a
20
So the total sales for the first 2o years is $1,625,000
152,500
1, 625, 000
9.3 – Geometric Sequences and Series
1, 4, 7, 10, 13 9, 1, 6.2, 6.6, 7, 7.4
3, 6 35 12 27.2
3 9
Arithmetic Sequences ADD To get next term
Arithmetic Series Sum of Terms 2, 4, 8, 16, 32 9, 1, 1/ 4, 1/16, 1/ 64 62 20 / 3 85 / 64 9.75
Geometric Sequences MULTIPLY To get next term
Geometric Series Sum of Terms
Vocabulary of Sequences (Universal) a 1 First term a n nth term n number of terms S n sum of n terms r common ratio nth term of geometric sequence a n a r 1 sum of n terms of geometric sequ ence S n 1 r n 1 1
Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 2 3 – 2 vs. 9/2 – 3… not arithmetic 9 / 2 3 1.5
geometric 3 2 2, 3, 9 , 2 9 2 3 2 , 9 2 2 3 2 , 9 2 2 3 2 3 2 2, 3, 9 , 2 27 81 243 , 4 8 , 16
If a 1 1 , r 2 2 , find a .
9 3 a 1 First term a n nth term n number of terms S n sum of n terms r common ratio 1/2 x 9 NA 2/3 a n a r 1 x x 2 3 9 1 2 8 8 2 7 3 8 128 6561
Find two geometric means between –2 and 54 -2, ____, ____, 54 a 1 First term a n nth term n number of terms S n sum of n terms r common ratio -2 54 4 NA x a n 54 27 x 3 x 4 1 The two geometric means are 6 and -18, since –2, 6, -18 , 54 forms an geometric sequence
Find a 2 a if a 4 1 3 and r 2 3 -3, ____, ____, ____ a 2 a 4 Since r 2 ...
3 2, 4 , 3 8 9 8 9 10 9
Find a of 9 2, 2, 2 2,...
a 1 First term a n nth term n number of terms S n sum of n terms r common ratio x 2 9 NA r 2 2 2 2 2 a n x x 2 2 9 1 8 x 16 2 2
If a 5 32 2 and r 2, find a 2 ____, ____ , ____,____ ,32 2 a 1 First term a n nth term n number of terms x 32 2 5 S n r sum of n terms NA common ratio a n 32 2 32 2 x x 5 1 4 2 32 2 4 x 8 2 x
4 *** Insert one geometric mean between ¼ and 4*** *** denotes trick question 1 4 r 3 1 a 1 1 ,____,4 4 First term a n nth term n number of terms S n sum of n terms r common ratio a n a r 1 1 r 2 4 16 r 2 4 r 1/4 4 3 NA x 1 4 1 4 , 1, 4
1 Find S of 7 2 S n 4 1 8 ...
a 1 First term a n nth term n number of terms 1 r n 1 1 S n sum of n terms r common ratio x 1 2 1 1 1 2 1 1 2 1 1 2 1/2 NA 7 1 63 64 x r 1 4 1 2 1 8 1 1 2 4
Section 12.3 – Infinite Series
1, 4, 7, 10, 13, ….
3, 7, 11, …, 51 Infinite Arithmetic Finite Arithmetic 1, 2, 4, …, 64 1, 2, 4, 8, … Finite Geometric Infinite Geometric r > 1 r < -1 1 1 1 3,1, , , 3 9 27 ...
Infinite Geometric -1 < r < 1 No Sum S n S n n 2 1 a 1 n r 1 a n 1 No Sum S a 1
Find the sum, if possible: 1 2 1 r 2 1 1 4 1 1 2 2 1 4 1 8 ...
1 Yes S a 1 1 1 1 2 2
Find the sum, if possible: 2 2 r 8 2 2 16 2 8 2 2 NO SUM ...
1 No
Find the sum, if possible: 2 3 3 1 6 1 12 ...
1 r 3 2 3 1 6 1 1 2 3 S a 1 2 1 3 1 2 4 3 1 Yes
Find the sum, if possible: 2 7 4 7 8 r 7 2 7 4 2 7 7 8 7 ...
1 No NO SUM
Find the sum, if possible: 10 5 2 ...
5 r 5 10 2 5 1 2 1 Yes S a 1 10 1 1 2 20
The Bouncing Ball Problem – Version A A ball is dropped from a height of 50 feet. It rebounds 4/5 of it’s height, and continues this pattern until it stops. How far does the ball travel?
50 40 40 32 32 32/5 32/5 S 50 1 4 5 40 1 4 5 45 0
The Bouncing Ball Problem – Version B A ball is thrown 100 feet into the air. It rebounds 3/4 of it’s height, and continues this pattern until it stops. How far does the ball travel?
100 75 225/4 100 75 225/4 S 100 1 3 4 10 1 0 3 4 80 0
Sigma Notation
UPPER BOUND (NUMBER) SIGMA (SUM OF TERMS) B a n NTH TERM (SEQUENCE) LOWER BOUND (NUMBER)
4 j 1 j 2 2 2 2 3 2 4 2 18 7 a 4 2 2 44 4 n 0 0.5
2 n 0.5
33.5
2 0 0.5
2 1 0.5
2 2 0.5
2 3 0.5
2 4
b 0 6 3 6 3 0 S a 1 3 1 6 6 1 3 5 15 2 ...
x 2 3 7 2x 1 S n 2 n 2 a 1 a n 2 3 2 ...
1 15 47 527 2 1
1 9 b 4 4b 3 4 3 S n n 2 a 1 a n 1 9 2 3 3 1 19 79 784 4 3
Rewrite using sigma notation: 3 + 6 + 9 + 12 Arithmetic, d= 3 a n a 1
a n
a n 3n 4 n 1 3n
Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1 Geometric, r = ½ a n a n 16 1 5 n 1 16 1
Rewrite using sigma notation: 19 + 18 + 16 + 12 + 4 Not Arithmetic, Not Geometric 19 + 18 + 16 + 12 + 4 -1 -2 -4 -8 a n 20 2 5 n 1 20 2
Rewrite the following using sigma notation: 3 5 9 10 27 15 ...
Numerator is geometric, r = 3 Denominator is arithmetic d= 5 NUMERATOR: 3 DENOMINATOR: a n a n n 1 a n 5n SIGMA NOTATION: n 1 5n n 1