Download Sequences and Series

Arithmetic Sequences and Series
I.
Arithmetic Sequence
Sequences of numbers that follow a pattern of adding a fixed number from one
term to the next are called arithmetic sequences.
Definition
A sequence with general term
an+1 = an + d
is called an arithmetic sequence.
an = nth term and d = common
difference
Definition
The nth or general term of an arithmetic sequence is given by
an = a1 + (n - 1)d
A1 = 1st term
Theorem
The sum of an arithmetic series is
Sn 
n
(a1  an )
2
This can be proven but let’s just convince ourselves that it works.
It is easy to determine the sum of the following arithmetic sequence:
3 + 5 + 7 = 15 This is an arithmetic series with common difference 2
Now let’s use the formula
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We have a1 = 3, an = 15, d = 2
Sn 
3
(3  7)  15
2
1 – 2 Find the following sums
1. -2 + 1 + 4 + 7 + 10
2. 3 + 7 + 11 + 15 + ... + 35
3 – 4 Find the 15th term
3. -1,10,21,32,43,54,...
4. 3,0,-3,-6,-9,-12,...
5. Application
Suppose that you play black jack at Harrah's on June 1 and lose $1,000. Tomorrow
you bet and lose $15 less. Each day you lose $15 less that your previous loss.
What will your losses be on the 30th of June? What will your total losses be for the
30 days of June?
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Geometric Sequences and Series
I.
Geometric Sequence
Sequences of numbers that follow a pattern of multiplying a fixed number from one
term to the next are called geometric sequences.
Definition
A sequence with general term
an+1 = an r
is called an geometric sequence.
an = nth term and r = common ratio
Definition
The nth or general term of an geometric sequence is given by
an = a rn-1 where a is the 1st term
Theorem
The sum of a geometric series is
n
1 r n
S n   a n  a
i 1
 1 r



This can be proven but let’s just convince ourselves that it works.
It is easy to determine the sum of the following geometric sequence:
3 + 6 + 12 = 21 This is a geometric series with common ratio 2
Now let’s use the formula
We have a = 3, an = 12, r = 2
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 1  23 
7
  3
S n  3
 3(7)  21
1
 1 2 
Find the 8th term
6. 2,6,18,54, ...
7. 27,9,3,1,…..
8. 16,-8,4,-2,1,...
Find the following sums
9. 5 + 10 + 20 + 40 + 80
10. -3 + 6 - 12 + 24
11. 3 + 1 + 1/3 + 1/9
12. Application
The parents of a 9-yr old boy have agreed to deposit $10 their sons bank account on
his 10th birthday and to double the size of their deposit every year thereafter until
his 18th birthday. How much will they have to deposit on his 19th birthday? How
much will they have deposited by his 18th birthday?
Exercises: If the series is arithmetic or geometric use the appropriate formulas to find the
7th term and the sum of the first seven terms. Must show work.
13. 1 + 4 + 8 + 13 + …..
14. -1 + 1 + 3 + 5 + …..
15. 5 + 15 + 45 +.... + 3645
16. 3 - 6 + 12 - .... – 96
1 3 1 9
17.    ……
2 8 4 64
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