Transcript 5-1 Exponential Rules

Arithmetic Progressions
“AP’s”
&
“GP’s”
Geometric Progressions
Arithmetic Progression is a sequence of numbers such
that the difference of any two successive members of the
sequence is a constant.
Example : 1 , 5 , 9 , 13
5 – 1 = 9 -5 = 13 – 9 This will help us fill in blanks in an AP
tn  a  (n  1)d
a  first term
n  # of terms in sequence
d= common difference
t n  nth term
Example
What is the 32nd term of 5 , 11 , 17 , 23 ….
tn  a  (n  1)d
t32  5  (32  1)6
Example
Fill in the blanks on the AP:
__ 5 __ __ __ 3
(5  d) (5  2d) (5  3d) (5  4d)
2 Ways to solve:
- Solve using the “d” value and an equation
- Insert Arithmetic Means
5+4d=3
Example
Which term of 4 , 16 , 28 , …… is 328?
tn  a  (n  1)d
Geometric Progression is a sequence of
numbers where each term after the first is found by
multiplying the previous one by a fixed non-zero
number called the common ratio.
1, 5, 25, 125,….. Where 1/5=5/25=25/125
tn  ar
n 1
Tn  nth term
a  First term
n  term number
r= common ratio
Example
What is the 6th term of 3/5 , 3 , 15 , 75 ….
tn  ar
n 1
Example
Fill in the blanks on the GP:
__ 4 __ __ __ 64
(4r) (4r2 ) (4r3 ) (4r4 )
- Create an equation with the common ratio
(4r 4 )  64
Example
Find the sum of the first 26 terms of the AP.
-5 , -1 , 3 , 7 ,……
We are missing tn
tn  a  (n  1)d
n(a  t n )
Sn 
2
Example
Find the sum of:
1
3

2k 2

k 2 
40
-Substitute 2 for k to solve for “a”
-Subtract 2 from 40 and add 1 to find the number of terms.
-Substitute 40 for k to solve for tn
-Find “d” by finding the second term and subtract from first
term.
Example
Solve: Find the sum of the following
8
2
k 1
k 1
Sn  ?
a 1
n  10
r 2
Example
Find all the values of x so that
2x  1,5x  4,4x  4
Are consecutive terms of an GP.
- Set up 2 equations, any ideas?
- 1st term/2nd term=2nd term/3rd term (GP’s have a ratio)
-
1 ,3 ,9
1/3 = 3/9
Example
Insert two numbers between 2 and 20 so that the first three
numbers form a GP while the last three numbers form an
AP
2
___ ___
(2r) (2r2 )
20
Use the AP:
    2r    2r 
20- 2r
2
2
Group Problem
The 3rd term of an AP is 6. The 2nd, 4th and 7th terms of the
AP form the first three terms of a GP. Find the first term of
the AP
Homework Assignment # 6