Download 13.1 Sequences

MAC1147
12.1 Sequences
A sequence is a function whose domain is the set of positive integers (natural numbers); that is, the terms are found by
calculating f(1), f(2), f(3) … f(n). Subscripted letters are usually used to represent the terms; e.g. a1 = the first term, an is
called the general term and represents all subsequent terms in the sequence. The three dots (…) is called an ellipsis and
indicates that the pattern continues infinitely. When a formula for the nth term, general term, is know, rather than writing
out the terms of the sequence, the entire sequence can be represented by placing braces around the formula for the nth
term.
Find the first five terms in the sequence:
1.
n

1  
a n     1   
n  

Write the nth term of the sequence suggested by the pattern:
2. 5, 6, 7, 8, …
3. 2, -4, 8, -16, …
n 1
Note:  1 acts as a sign alternator.
4.
1 2 3 4
, , , ,
2 3 4 5
5.
3
3
3, , 27,
4
16
Recursive formulas define a sequence by assigning values to the first or first few term(s) and specifying the nth
with a formula that involves one or more of the terms preceding it.
Find the next four terms in the sequence:
6. a1 = 5, an = an-1 + 2, n ≥ 2
7. b1 = 1, b2 = 3, bn = nbn-2 + bn-1, n ≥ 3
The factorial symbol n! is defined as 0! = 1, 1! = 1, n! = n(n – 1) . … . 3 . 2 . 2, n ≥ 2 [5! = 5(4)(3)(2)(1) =
n
120]. The summation symbol
a
k 1
k
means to add a1 + a2 + … + an. The index integer k tells where to start
the sum. The n on top tells where to end the sum.
Find the sum of each sequence:
10
8.
3
5
5
9.
k 1
k
5
k 0

Write each series in expanded form without summation form:
4
10.

k 1
 2 
k 1
k
Express the sum using summation notation:
1 1 1
1
1 1 1 1 1
11. Sn      n
12. S6  1     
2 4 8
2
2 3 4 5 6
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MAC1147
TRY THESE (12.1)
1
2
1
4
1
6
1
8
1. Write the nth term of the sequence suggested by the pattern: 1, , 3, ,5, ,7, ,
an  1
2
2. Write the first five terms: a1  2, an 
4
3. Find the sum:
  1 
k
3k
 1 
1
31   1 32   1 33   1 34  3  9  27  81  60
2
k 1
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3
4
MAC1147
12.2 Arithmetic Sequences
An arithmetic sequence (sometimes called an arithmetic progression) is one in which the difference between successive
terms of the sequence is always the same number. It can be defined recursively as a1 = a, an = an-1 + d, where a and d are
real numbers. The number a is the first term and d is the common difference. A sequence can be shown to be arithmetic
if subtracting successive terms yields a constant. For an arithmetic sequence {an} whose first term is a and whose
common difference is d, the nth term is determined by the formula an = a + (n – 1)d.
Determine if the following sequences are arithmetic:
1. 5, 9, 13, 17, …
2.
{sn} = {5n + 9}
3.
Find the 50th term of 5, 9, 13, 17, …
4.
Find the 20th term of {sn} = {5n + 9}
The sum Sn of the first n terms of an arithmetic sequence {an} whose first term is a and whose common difference is d is
determined by the formula Sn 
n
n
 2a   n  1 d    a  an  .
2
2
Find the sum of the following arithmetic series:
5. 3 + 6 + 9 + … + 180
7.
6. {5n + 9}
Find the sum of the first 52 terms of an arithmetic series if the first term is 23 and d = -2.
Find a recursive formula for the sequence:
8. Find the recursive formula for the sequence whose 27th term is 13 and 32nd term is -22.
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MAC1147
TRY THESE (12.2)
2 n
1. Find the first four terms of    (Show terms in fractional form).
3
4
2. What is the seventh term of the series 2 5, 4 5, 6 5,
3. Find the sum of 1 + 3 + 5 + … + 59
4. Find the recursive formula for the sequence whose 12th term is 4 and 18th term is 28.
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MAC1147
12.3 Geometric Sequences
An geometric sequence (sometimes called an geometric progression) is one in which the ratio of successive terms of the
sequence is always the same nonzero number. It can be defined recursively as a1 = a, an = r an-1 or an/ an-1 = r, n > 0,
where a and r ≠ 0 are real numbers. The number a is the first term and r is the common ratio. A sequence can be shown
to be geometric if dividing successive terms yields a constant. For a geometric sequence {an} whose first term is a and
whose common ratio is r, the nth term is determined by the formula an = arn-1, r ≠ 0.
Determine if the following sequences are arithmetic or geometric:
1. 2, 6, 8, 10 …
 
n
2
2.
{sn} = 3
3.
1 1
Find the 7th term of 1, , , ...
2 4
4.
Find the 20th term of the geometric sequence in which a = 7 and r = 3
The sum Sn of the first n terms of an geometric sequence {an} whose first term is a and whose common ratio is r
is determined by the formula Sn  a 1  r , r  0,1 . An infinite sum of the form ar + ar2 + … + arn is called an
n
1 r
infinite geometric series and is denoted by

 ar
k 1
. If the finite sum approaches a number L as n  , then L
k 1
is called the sum of the infinite geometric series: L =

 ar
k 1
k 1

a .
1 r
Find the sum of the following geometric series:
n
 2

6. Sn   
k 1  3 
5.
1 – 5 + 25 - 125 + 625
7.
Find the sum of the infinite series: 2 
1 1
 
2 8
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k
MAC1147
TRY THESE (12.3)
1.
 2n 
Find the common ratio and write out the first four terms:  n1 
3 
2.
Find the 10th term of: -1, 2, -4, …
3.
Find the sum 2 
6 18
 3

 ...  2  
5 25
5
n

4.
1
Find the sum of the infinite series  8  
k 1  3 
k 1
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