Unit- Sequences and Series
1
Algebra 2
Lesson/HW- Summation Notation
Name:____________________________________
Date:_____________________________________
SHOW ALL WORK:
Write each expression in expanded form and then find the sum:
4
(1)

n 1
5
2
n
 n
(2)
n 1
5
(3)
2
 2n

3
(4)
n3
2
 n  2
4
(6)
(7)

n 1
 1
10 
2

n 1
n2

 2n  3
n0
5
(5)
 nn  1
n

(8)

n0
n 2 
 4
 
 5
n
2
Algebra 2
Lesson- Arithmetic Sequences
Name:____________________________________
Date:_____________________________________
Objective:
To learn what an arithmetic sequence is & how to find the sum of an arithmetic sequence
DO NOW:
Find a rule that is representative of the following series:
0+3+8+15+24
__________________________________________________________________________________________
What is an Arithmetic Sequence? How do I identify one?
an  dn  c where d is the difference in consecutive terms of the
sequence and c  a1  d
The nth term of an arithmetic sequence:
Ex 1: Find the nth term of: 7, 11, 15, 19,…
Ex 2: The fourth term of an arithmetic sequence is 20, and the thirteenth term is 65. State the general rule &
write the first several terms of this sequence.
Sum of Arithmetic Sequence:
S
n
(a1  an )
2
150
Ex 3: Find the sum of the integers from 1 to 100.
Ex 4: Find the sum:
 (11n  6)
n 1
Ex 5: Verify:
S  1  3  5  ...  (2n  1)  n2
3
Algebra 2
Lesson- Geometric Sequences
Name:____________________________________
Date:_____________________________________
Objective:
To learn what a geometric sequence is & how to find the sum of an geometric sequence
DO NOW:
Find a rule that is representative of the following series:
2+5+10+17+26
__________________________________________________________________________________________
What is an Geometric Sequence? How do I identify one?
The nth term of an arithmetic sequence:
a n  a1r n 1 where r is the common ratio of consecutive terms of
the sequence. Every geometric sequence in the form of:
a1 , a2 , a3 , a4 , a5 , a6 ,..., an can be written as:
a1 , a1r , a1r 2 , a1r 3 , a1r 4 , a1r 5 ,..., a1r n1
Ex 1: Find the nth term of: 2, 4, 8, 16, 32,…
Ex 2: Find the 15th term of a sequence whose first term is 20 and whose common ratio is 1.05.
Ex 3: Find the 12th term of: 5, 15, 45,…
Ex 4: The fourth term of a geometric sequence is 125, and the 10th term is
125
. Find the 14th term.
64
SUMS
1 r n 

Finite Geometric Sequence: S  a1 
 1 r 
Infinite Geometric Sequence: S   a1r n1 
 1 
Ex 5: Find the sum: 10 
 2 
i 0
Ex 6: Find the sum: 4, 4(0.6), 4(0.6) 2 , 4(0.6)3 ,...,4(0.6) n1
10

n1
a1
1 r
i
4
Algebra 2
Lesson/HW- Writing in Sigma Notation
Name:______________________________
Date:_______________________________
Objective:

find the general term of a series and write a series in sigma notation
Find the general term of a sequence whose first four terms are given:
(a) 7, 14, 21, 28, …
(b)
3 7 11 15
, , ,
, ...
3 4 5 6
(c)
1 1
1 1
 , , ,
, ...
1 4
9 16
Writing a Series in Sigma Notation:
(1)
5 + 10 + 15 + 20 + 25 + 30
(2)
1
1 1
1
  ... 
4 9
100
(3)
15 + 24 + 35 + 48 + … + 143
(4)
1
3 5 7
 
2 3 4
(5)
1
(6)
-4 + 16 – 64 + 256
(7)
1 3 7 15
  
2 4 8 16
(8)
1
(9)
2 – 1 – 4 – 7 – … – 19
(10) 1 
(11)
2
(13)
1
1
1


10 100 1000
1 1 1 1 1
   
2 3 4 5 6
3 4 5 6
  
2 3 4 5
(15) 5 + 7 + 9 + 11 + 13
2 4 8 16
 

3 9 27 81
1 1
1


5 25 125
(12) 9 + 12 + 15 + 18 + … + 30
(14) -1 + 1 – 1 + 1
(16) 1 
1 1 1 1 1
   
3 5 7 9 11
5
Algebra 2
Review- Sequences and Series Test
Name_____________________________________
Date:_____________________________________
Answer each of the following neatly and completely on separate paper.
Let a1, a2, a3, …,an, … be an arithmetic sequence. Find the indicated quantities.
1.
a1 = -5, d = 4, a2 = ?, a3 = ?, a4 = ?
2.
a1 = -3, d = 3, a2 = ?, a3 = ?, a4 = ?
3.
a1 = -3, d = 5, a15 = ?, S11 = ?
4.
a1 = 1, a2 = 5, S21 = ?
5.
a1 = 7, a2 = 5, a15 = ?
6.
a1 = 1/3, a2 = ½ , a11 = ?, S11 = ?
7.
a1 = 3, a20 = 117, d = ?, a101 = ?
8.
a1 = -12, a40 = 22, S40 = ?
9.
Find g(1) + g(2) + g(3) + … + g(51) if
g(t) = 5 - t
10.
Find the sum of all the even integers
between 21 and 135.
12.
an = n + 3
Write the first four terms of the given sequences
11.
an = n – 2
13.
n 1
an =
n 1
14.
1
an = 1  
n

15.
an = (-2)n+1
16.
an =
n
 1n1
n2
Write the first five terms in each sequence
1
1 
1  n 
3  10 
17.
an = (-1)n+1n2
18.
an =
19.
an = (- ½ )n-1
20.
a1 = 7, an = an-1 – 4, n > 2
21.
a1 = 4, an = ¼ an-1, n > 2
6
Find the general term of the sequence for which the first four terms are given:
22.
4, 5, 6, 7, …
23.
3, 6, 9, 12, …
24.
1 1 1 1
, , , ,...
2 3 4 5
25.
1, -1, 1, -1, …
26.
–2, 4, -8, 16, ….
27.
1, -3, 5, -7, ….
28.
x,
x2 x3 x4
, , ,...
2 2 2
29.
x, -x3, x5, -x7, ….
32.
1
 

k 1  3 
Write each series in expanded form and then find the sum
(1) k 1

k
k 1
30.
6
34.
 (1)
k 1
5
3
5
31.
1

k
k 1 10
(2 k 1 )

k
k 1
k
35.
4
33.
36.
1 k 1
x

k 1 k
 (1)
k
k 1
3
4
k 1
k
(1) k 1 k
x

k
k 1
5
37.
Write each series using summation notation with the summing index k starting at k=1
38.
12 + 22 + 32 + 42
40.
1
1
1
1
 2  ...  2
2
2
3
n
39.
1 1
1
1
 2  3  ...  n
2 2
2
2
41.
1 – 4 + 9 – 16 + 25 – 36 + 49 – 64
7
Find the indicated quantities in each geometric sequence
2
, a2 = ?, a3 = ?, a4 = ?
3
42.
a1 = -6, r = - ½, a2 = ?, a3 = ?, a4 = ?
43.
a1 = 12, r =
44.
a1 = 100, a6 = 1, r = ?
45.
a1 = 5, r = -2, S10 = ?
46.
a1 = 81, r =
1
, a10 = ?
3
47.
a1 = 3, a7 = 2,187, r = 3, S7 = ?
Find the sum of each infinite geometric series that has a sum
1
48.
3 + 1 + + ….
49.
3
50.
2–
1
1
+ +…
2
8
51.
2+4+8+…
5+
5
5
+ +…
2
4
8