1. Pre-Calculus 11 Notes
Mr. Rodgers

2. Chapter 1 Sequences and Series

3. Chapter 1.1 Arithmetic Sequences
An arithmetic sequence is an ordered list of terms, where the difference between consecutive terms is constant, or, when the same value or variable is added to each term to create the next term. This constant is called the common difference (d). In order to find the common difference, you subtract the first term from the second term of any two consecutive terms ( ex. d = t2 - t1 ).
The formula for an arithmetic sequence is:
tn = t1 + (n-1)d where t1 is the first term of the sequence
n is the number of terms
d is the common difference
tn is the general term or nth term

4. Example:
A performing arts group wants to hire an events leader. The person will be paid $12 for the first hour of work, $19 for two hours of work, $26 for three hours of work. What will the person get paid for 6 h of work?
Solution:
t1 = 12
t2 = Subtract the first term from the second term that will result to a common
t3 = 26 … difference of 7.
t1 = 12, d = 7
tn = t1 + (n-1)d
tn = 12 + (n-1)7
tn = n-7
tn = 7n + 5
For 6 h of work, determine t6
tn= 7n + 5
t6= 7(6) + 5
t6=
t6= For 6 h of work, the person will be paid $47.

5. 1.2 Arithmetic Series
An arithmetic series are a sum of terms that form an arithmetic sequence.
Sn represents the sum of the first n terms of a series.
The following is an arithmetic sequence: 1,3,5,7,9… In this case, S5 means the sum of the first 5 terms.
The formulas are:
Sn= n [2t1 + (n-1)d] Sn= n (t1 + tn)
Where t1 is the first term Where t1 is the first term
n is the number of terms n is the number of terms
d is the common difference tn is the nth term
Sn is the sum of the first n terms Sn is the sum of the first n terms

6. Example 1:
Find the sum of the first 60 terms of the series n = 60, t1= 5, d = 3
S60 = 60 (2(5) + 59 (3)) 2
= 30 (10+177)
= 30 (187)
S60 = 5610
Example 2:
Find the sum of the arithmetic series t1 = 5, tn= 201, d = 4
tn= t1 + (n-1)d
201 = 5 + (n-1) Sn= n [2t1 + (n-1)d]
196 = 4(n-1)
49 = n S50 = 50 ( )
50 = n
= 25 (210)
S50 = 5250

7. 1.3 Geometric Series
A geometric sequence is a sequence when the ratio of consecutive terms is constant. The common ratio, r, can be found by taking any term, and dividing that term by the preceding term
( ex. r = t2 ÷ t1 ).
The formula is:
tn = t1r n-1 where t1 is the first term of the sequence
n is the number of terms
r is the common ratio
tn is the general term or nth term
the ratio may be positive or negative
for example, in the sequence 2, 4, 8, 16,..., the common ratio is 2

8. Example 1:
In the geometric sequence determine the general term tn and the seventh term.
32, 16, 8, … t1= 32, r = 1 2
t7= 32 (1) 7-1
t7= 32 (1) t7= 32 (1) t7= t7= 1
Example 2:
Bacteria reproduce by dividing in two so that one cell gives rise to 2, then 4, then 8 cells, and so on, producing a geometric sequence. Suppose there were 10 bacteria in a sample. Determine the general term and state the values of t1 and r.
t1= 10, t2= 20, t3= Divide the second term by the first term to find a common ratio of 2.
tn = t1r n t1= 10 and r = 2
tn = (10)(2) n-1
The general term of the sequence is tn = 10(2) n-1.

9. 1.4 Geometric Series
A geometric series is the sum of the terms in a geometric sequence. We use the symbol Sn to represent the sum of the first n terms of a geometric sequence.
For example, is a geometric series.
The formula is:
Sn= t1(r n- 1) , r ≠ 1 where t1 is the first term of the series
r-1
n is the number of terms
r is the common ratio
Sn is the sum of the first n terms

10. Example 1:
Determine the sum of the first 10 terms of the geometric series.
… t1 = 4, r = 3, n = 10
Sn= t1(r n- 1)
r-1
S10= 4(310-1) S10= 4(59 048) S10=
Example 2:
In a geometric sequence, the fifth term is 1024 and the common ratio is 4. Find the sum of the first seven terms of this series t5= 1024, r = 4, n = 7 Find Sn
tn = t1r n Sn= t1(r n- 1)
1024 = (256)t divide both sides by 256
4 = t S7 = 4(47-1)
4-1
S7 = 21844

11. 1.5 Infinite Geometric Series
The formula for an infinite geometric series is:
S∞ = t < r < 1 where t1 is the first term of the series
1-r
r is the common ratio
Sn represents the sum of an infinite number of terms
An infinite geometric series can be either convergent or divergent.
A convergent series is when the sequence of partial sums approaches a fixed value, or when r is within the range of -1 < r < 1
( ex … )
A divergent series is when the sequence of partial sums does not approach a fixed value, or when r is out of the range of -1 < r < 1.
( ex … )

12. Example:
Decide whether each infinite geometric series is convergent or divergent. State the sum of the series if it exists.
… b) …
t1 = 1, r = t1 = 2, r = -2 3
Since -1 < r < 1, the series is convergent Since r < - 1, the series is divergent
and has no sum.
S∞ = t1
1-r
S∞ =
1 - (-1)
( 3)
S∞ = 1
(4)
(3)
S∞ = (1) S∞ = 3