This document provides an overview of honors level topics in module 6 including exponential and logarithmic inequalities, non-arithmetic and non-geometric sequences and series, recursive and explicit formulas for sequences, and sigma notation for sums. Examples are given of solving exponential and logarithmic inequalities, writing terms of sequences using recursive and explicit formulas, and evaluating sums using sigma notation. Formulas and steps for working with exponents, logarithms, sequences and series are reviewed.

1. MATH 3 - MODULE 6
Honors Topics

2. Exponential and Logarithmic Inequalities
• Exponential inequality rules:
• Logarithmic inequality rules:
If the bases of the exponential inequality are not the same, you must
“log both side” to get the variable out of the exponent.
, thenx y
b b x y  , thenx y
b b x y 
If 1, then log 0bn n  If 0 < 1, then log 0bn n 
log log
log log
log log
if 1 and if 0 < 1
log log
x
x
b c
b c
x b c
c c
x b x b
b b



   
If log log , thenb bn c n c  If log log , thenb bn c n c 
**Always check
solutions for
logarithms- must have
only positives after
the log

3. Examples of
Exponential and Logarithmic Inequalities
• Solve each inequality.
2 1
2 1
1
1
3
0
3 x x
x x
x
x

 
 


log5.2 log 4
log5.2 log 4
log 4
You are by positive
log
5.
5.2
0.84 )
2 4
(
x
x
x
x
x approx




 
log0.47 log8.1
log0.47 log8.1
log8.1
You are by n
0
egative
log0.
.47 8.
47
2.77( )
1x
x
x
x
x approx


 
 

3 3log ( 4) log (3 )
4 3
4 2
2 2
x x
x x
x
x x
 



 
 2 2log (5 2) log ( 4
5 2 4
4 6
)
3
2
x x
x
x
x
x
  



 

4. Non-Arithmetic and Non-Geometric
Sequences & Series
• We studied arithmetic and geometric sequences and
series, but there are some sequences and series that are
neither arithmetic nor geometric.
• Sequences can be generated using any pattern of n, the
location and number of each term.
• generates the following terms. A table is a
good way to organize the terms.
*This sequence does not have a common difference or common ratio
2
2na n 
n 1 2 3 4 5 6
-1 2 7 14 23 34na
2
3 3 2a  2
2 2 2a  2
1 1 2a   2
4 4 2a   2
5 5 2a   2
6 6 2a  

5. Terms of Sequences
• Find the first 4 terms of each sequence.
Terms: 0, 1/5, 1/3, 3/7
Terms: 5, 7, 11, 19
*These are all explicit formulas, but can you use recursive?
1
3
n
n
a
n



n 1 2 3 4
0 1/5 1/3 3/71
3
n
n
a
n



2 3n
na  
n 1 2 3 4
5 7 11 192 3n
na  

6. Examples of Recursive Formulas
• Find the first 4 terms of each sequence.
Terms: -4, -7, -13, -25
Terms: 5, 7, 11, 19
Now that you generated terms, can you write the formulas?
n 1 2 3 4
-4 -7 -13 -25
na
 
2
1 1
1
given
2
n na a a  n 1 2 3 4
1/2 1/4 1/16 1/256
1 12 1 if 4n na a a   
na

7. Write Explicit Formulas
• You may want to organize the terms in a table to compare
the terms to the values of n.
• Do you add to n? Subtract? Multiply? Divide? Square it?
• Write the explicit formula for the apparent nth term of the
sequence.
• 1, 4, 7, 10, 13, …
Formula:
• 2, 5, 10, 17, 26
Formula:
n 1 2 3 4 5
1 4 7 10 13na
3 2na n 
n 1 2 3 4 5
2 5 10 17 26na
2
1na n 

8. Sigma Notation
• Find the indicated sum.
4
1
1 1 1 1
1 2 3 4
12 6 4 3 25
12 1
1
2
i i
  
  


6
2
20 30 40 50 60
200
10
k
k

   

