The document provides information on arithmetic and geometric sequences. It gives the formulas for finding the nth term and sum of terms for both arithmetic and geometric sequences. Examples are provided of identifying sequences as arithmetic or geometric and calculating sequence terms and sums. Steps are shown to find missing terms, the first term given later terms, and common ratios of geometric sequences.

1. 63
Find S of 19, 13, 7,...
  
1
a First term

n
a nth term

n
S sum of n terms

n number of terms

d common difference

-19
63
??
x
6
 
n 1
a a n 1 d
  
  
?? 19 6 1
?? 353
3 6
  


353
 
n 1 n
n
S a a
2
 
 
63
63
3 3
S
2
19 5

 
63 1 1
S 052


2. 16 1
Find a if a 1.5 and d 0.5
 
Try this one:
1
a First term

n
a nth term

n
S sum of n terms

n number of terms

d common difference

1.5
16
x
NA
0.5
 
n 1
a a n 1 d
  
 
16 1.5 0.
a 16 5
1
  
16
a 9


3. n 1
Find n if a 633, a 9, and d 24
  
1
a First term

n
a nth term

n
S sum of n terms

n number of terms

d common difference

9
x
633
NA
24
 
n 1
a a n 1 d
  
 
633 9 2
1
x 4
  
633 9 2 24
4x
  
X = 27

4. 1 29
Find d if a 6 and a 20
  
1
a First term

n
a nth term

n
S sum of n terms

n number of terms

d common difference

-6
29
20
NA
x
 
n 1
a a n 1 d
  
 
1
20 6 29 x
  

26 28x

13
x
14


5. Find two arithmetic means between –4 and 5
-4, ____, ____, 5
1
a First term

n
a nth term

n
S sum of n terms

n number of terms

d common difference

-4
4
5
NA
x
 
n 1
a a n 1 d
  
  
1
5 4 4 x
  

x 3

The two arithmetic means are –1 and 2, since –4, -1, 2, 5
forms an arithmetic sequence

6. Find three arithmetic means between 1 and 4
1, ____, ____, ____, 4
1
a First term

n
a nth term

n
S sum of n terms

n number of terms

d common difference

1
5
4
NA
x
 
n 1
a a n 1 d
  
  
4 1 x
1
5
  
3
x
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

7. Find n for the series in which 1 n
a 5, d 3, S 440
  
1
a First term

n
a nth term

n
S sum of n terms

n number of terms

d common difference

5
x
y
440
3
 
n 1
a a n 1 d
  
 
n 1 n
n
S a a
2
 
 
y 5 3
1
x
  
 
x
40 y
4
2
5
 
 
 
1
2
x
440 5 5 x 3
   
 
x 7 x
440
2
3


 
880 x 7 3x
 
2
0 3x 7x 880
  
X = 16
Graph on positive window

8. Geometric Sequences and Series

9. 1, 4, 7,10,13
9,1, 7, 15
6.2, 6.6, 7, 7.4
, 3, 6
 
    
Arithmetic Sequences
ADD
To get next term
2, 4, 8,16, 32
9, 3,1, 1/3
1,1/ 4,1/16,1/ 64
, 2.5 , 6.25
 
  
Geometric Sequences
MULTIPLY
To get next term
Arithmetic Series
Sum of Terms
35
12
27.2
3 9

 
Geometric Series
Sum of Terms
62
20/3
85/ 64
9.75

10. Vocabulary of Sequences (Universal)
1
a First term

n
a nth term

n
S sum of n terms

n number of terms

r commonratio

 
n 1
n 1
n
1
n
nth term of geometric sequence
sum of n terms of geometric sequ
a a r
a r 1
S
r 1
ence

 
 

 




11. Find the next three terms of 2, 3, 9/2, ___, ___, ___
3 – 2 vs. 9/2 – 3… not arithmetic
3 9/ 2 3
1.5 geometric r
2 3 2
    
3 3 3 3 3 3
2 2 2
9
2, 3, , , ,
2
9 9 9
2 2 2 2 2 2
     
9
2, 3, , ,
27 81 243
4 8
,
2 16

12. 1 9
1 2
If a ,r , find a .
2 3
 
1
a First term

n
a nth term

n
S sum of n terms

n number of terms

r common ratio

1/2
x
9
NA
2/3
n 1
n 1
a a r 

9 1
1 2
x
2 3

  
   
  
8
8
2
x
2 3


7
8
2
3

128
6561


13. Find two geometric means between –2 and 54
-2, ____, ____, 54
1
a First term

n
a nth term

n
S sum of n terms

n number of terms

r commonratio

-2
54
4
NA
x
n 1
n 1
a a r 

  
1
4
54 2 x



3
27 x
 
3 x
 
The two geometric means are 6 and -18, since –2, 6, -18, 54
forms an geometric sequence