Sequences

1
Sequences
For the sequence 6, 11, 16, , 731 How many
terms?
What sort of question?
nth term of arithmetric sequence
u1
d
5
6
un
731
731
6 5(n-1)
n -1
(731-6)/5 145 n 146
2
Sequences
If I put 400 in a bank account on my 16th
birthday and get 5 interest per year. How much
money would I have on my 41st birthday?
What sort of question?
nth term of geometric sequence
u1
r
1.05
400
n
25
un
400 x 1.05 24
un
1290.04
3
Sequences
For the sequence 7, 9, 11, 13, .. What is the
sum of the first 50 terms?
What sort of question?
Sum of terms in arithmetic sequence
50
u1
d
2
7
n
S50
25(14 49 x 2)
S50
25 x 112 2800
4
Sequences
The sequence 2, 4a, 8a2, . Find an expression
for the sum of the first 10 terms
What sort of question?
sum terms of geometric sequence
u1
r
2a
2
n
10
S10
2((2a)10-1)/(2a-1)
S10
(211a10-2)/(2a-1)
5
Logarithms and Exponentials

• Learning Objective
• Revise

6

The Laws of Logs

• Laws of logs
• log a log b log ab
• Example log 3 log 5 log 15
• log a - log b log (a/b)
• Example log 21 log 7 log 3
• log a k k log a
• Example log 35 5 log 3

7

Logarithms Identities

• Given a log x and b log y, show
• log (x / y3) a 3b
• use the Laws of Logs

• log a - log b log a/b

log (x/y3) log x - log y3

• log a k k log a
• log (x/y3) log x 3 log y
• log (x/y3) a 3b QED

8
Log Formula
Logs and Exponentials are inverses
9
Solving Equations by Inverses
Inverse of is -
x 6 10
x 10 - 6
Inverse of is
7y 42
y 42 7
Inverse of log is 10
log x 3
x 103
10x 5
Inverse of 10 is log
x log 5
10

Log and Exponential Equations 1

• Solve
• log (y2) 5

1
2
10(y2) 5
log 10(y2) log 5
Inverse of 10 is log10
(y2) log 10 log 5
y2 log 5 y log (5) - 2
11

Log and Exponential Equations 2

• Solve
• log2 (5y2) 5

3
4
2(5y2) 5
log 2(5y2) log 5
Inverse of 2 is log2
(5y2) log 2 log 5
5y2 log 5/log 2 y ((log 5/log 2) - 2)/5
12

Logarithms
The old times by 1 method

• Solve
• log4 (3a2) 2

log4 4 1
log4 4
log4 (3a2) 2
X 1
log4 (3a2) log4 42
13

Natural Logarithms
ax ex lna

• Solve
• ln (y2) 5

5
6
e(y2) 5
ln e(y2) ln 5
Inverse of e is loge ln
(y2) ln e ln 5
y2 ln 5 y ln (5) - 2
14

Logarithms eg4

• Solve
• 33x2 - 1 7
• use the Laws of Logs

33x2 7 1 8
log 33x2 log 8
(3x2) log 3 log 8
3x2 log 8 log 3 3x2 1.893
3x 1.893 2 -0.107
x -0.107/3 -0.036
15

Logarithms Log Equations 3

• Solve
• log4 x log4 2x 3

• log a log b log ab

log4 4 1
log4 2x2 3
log4 4
log4 2x2 3
log4 2x2 log443