1 ARITHMETIC AND GEOMETRIC SERIES Standards 22 and 23 Sum of Infinite Geometric Series: Problems Sum...
15
ARITHMETIC AND GEOMETRIC SERIES Standards 22 and 23 Sum of Infinite Geometric Series: Problems Sum of Arithmetic Series: Problems Geometric Sequence: Problems Arithmetic Sequence: Problems Sum of Geometric Series: Problems END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights re served
-
Upload
grant-atkins -
Category
Documents
-
view
218 -
download
1
Transcript of 1 ARITHMETIC AND GEOMETRIC SERIES Standards 22 and 23 Sum of Infinite Geometric Series: Problems Sum...
1
ARITHMETIC AND GEOMETRIC SERIES
Standards 22 and 23
Sum of Infinite Geometric Series: Problems
Sum of Arithmetic Series: Problems
Geometric Sequence: Problems
Arithmetic Sequence: Problems
Sum of Geometric Series: Problems
END SHOWPRESENTATION CREATED BY SIMON PEREZ. All rights reserved
2
STANDARD 22:
Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.
STANDARD 23:
Students derive the summation formulas for arithmetic series and for both finite and infinite geometric series.
ALGEBRA II STANDARDS THIS LESSON AIMS:
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
3
ESTÁNDAR 22:
Los estudiantes encuentran términos generales y sumas de series aritméticas y de series geométricas finitas e infinitas.
ESTÁNDAR 23:
Los estudiantes derivan las fórmulas de suma para series aritméticas y series geométricas finitas e infinitas.
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
4
Standards 22 and 23
Find the indicated term in each arithmetic sequence.
1a for a = 6 , and d=4. 16
a = a + (n - 1)d n 1
n= 16
16a = 6 + (16-1)(4)
= 6 + 15(4)
= 6 + 60
= 66
a for 2, 6, 10, 14,…12
a = a + (n - 1)d n 1
n= 12
12a = 2 + (12-1)(4)
= 2 + 11(4)
= 2 + 44
= 46
d= 6-2=4
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
5
Standards 22 and 23
Find the missing terms in this arithmetic sequence.
6, __, __, __, -6a = a + (n - 1)d n 1
1st 5th
1a =65a = -6
n= 5
-6 = 6 + (5-1)d-6 = 6 + 4d
-6 -6
-12 = 4d
-12 = 4d4 4
d = -3
2a = 6-3
=3
3a = 3-3
=0
4a = 0-3
= -3
3 0 -3
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
6
Standards 22 and 23
Find the missing terms in this arithmetic sequence.
__, __, 12, __, 22a = a + (n - 1)d n 1
3rd 5th
3a =125a = 22
n= 5
2a = 2+5
=7
4a = 12+5
= 17
d = 22-122
= 5
22 = a + (5-1)(5)1
22 = a + (4)(5)1
22 = a + 201
-20 -20
1a = 2
7 172
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
7
Standards 22 and 23
Find S for this arithmetic series described.n
a = 14, a =210, n= 15. 1 n
S = (a + a )n 1 nn2
S = (14 + 210)15 215
S = (224)15 215
S = (7.5)(224)15
S = 168015
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
8
Standards 22 and 23
Find S for this arithmetic series described.n
S = (a + a )n 1 nn2
with and a = a + (n - 1)d n 1
S = (a + a + (n-1)d)n 1 1n2
S = (2a + (n-1)d)n 1n2
a = 18, n= 20, d = 3 1
S = (2(18) + (20-1)(3))20 20 2
S = (36 + (19)(3))20 20 2
S = (36 + 57)20 20 2
S = (93)20
20 2
S = (10)(93)20
S = 93020
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
9
Standards 22 and 23
Find the first three terms of this arithmetic series.
a = 5, a =61, S = 495. 1 n n
S = (a + a )n 1 nn2
495 = (5 + 61)n2
495 = (66)n2
495 = 33n33 33
n= 15
a = a + (n - 1)d n 1
61 = 5 + (15-1)d61 = 5 + 14d
-5 -5
56 = 14d
56 = 14d14 14
d = 4
2a = 5+4
=9
3a = 9+4
= 13
First let’s find n: Now let’s find d: Finally let’s find second and third terms:
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
10
Standards 22 and 23
Find the nth term of each geometric sequence.
a = -6, r=3, n=61
a = a r 1nn-1
a = (-6)(3) 66-1
a = (-6)(3) 65
a = (-6)(243) 6
a = -1458 6
a = 4, r= -2, n=83
First let’s find a1 4-2
a =2 = -2
-2 -2
a =1 = 1
Now using a = a r 1nn-1
a = (1)(-2) 88-1
a = (1)(-2) 87
a = (1)(-128) 8
a = -128 8
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
11
Standards 22 and 23
Find the nth term of this geometric sequence.
First let’s find a1
a =2
Now using a = a r 1nn-1
a = , r= , n=93 4 5
2 3
4 5 2 3
= (4)(3) (2)(5)
6 5
=
a =1
6 5 2 3
= (6)(3) (2)(5)
9 5
=
a = ( )( ) 99-1 9
5 2 3
a = ( )( ) 98 9
5 2 3
a = ( )( ) 9 9 5
256 6561
a =9 256 3645
a =9 2304 32805
.
...99
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
12
Standards 22 and 23
Find the sum of this geometric series.
7 + 21 + 63 + … to eight termsFirst find r:
21 7
r= = 3
a =7, r=3, n=81
S = a - a r1 1
n
1 – r n
S = 7 – (7)(3)8
1 - 3
8
S = 7 – (7)(6561)
-2
8
S = 7 – 45927
-2
8
S = -45920
-2
8
S = 229608
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
13
Standards 22 and 23
Find the sum of this geometric series.
3 + -6 + 12 + … to 7 termsFirst find r:
-6 3
r= = -2
a =3, r=-2, n=71
S = a - a r1 1
n
1 – r n
S = 3 – (3)(-2)7
1 - -2 7
S = 3 – (3)(-128)
3 7
S = 3 + 384
3 7
S = 3873
7
S = 1297
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
14
Standards 22 and 23
Find the sum of this infinite geometric series, if it exists.
a = , r= 1 2 3
1 2
Since |r|<1 it has a sum
S=a1
1 - r
S =
2 3
1 - 1 2
S =
2 3 1 2
= (2)(2) (3)(1)
4 3
=
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
15
Standards 22 and 23
Find the sum of this infinite geometric series, if it exists.
-5 + 15 + -45 + 135 +…
Lets find r first:
15-5
r= = -3
Since |-3| > 1 this sum does not exist
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
