Download Notes on finding the Nth term in sequences using factoring

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Lesson 1.4
Finding the nth Term
ExampleB
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'UI01t
of mono
Tenn
Value
- £eopoM ~rone
The pattern of numbers 3, 4, 5, 6, 7. 8, 9, ... is an example of a sequence of
numbers. The first term of the sequence has a vaI"ueof three, the second term has a
value of four, the third term has a value of five, and so on. Let's show this as a
table.
Term
1
2
3
4
5
6
..
20
Value
3
4
5
6
7
8
...
22
,
...
...
200
...
202
... n+2
3
4
5
6
-2
-1
0
1
2
3
...
...
20
-?-
...
...
200
-1-
...
...
5
6 ....
20
...
200
...
n
2
4
6
8
10
12
...
-?-
...
-1-
...
-7-
Term
1
Value
1
2
4
3
3
5
7
5
6
9
11
...
...
20
-?-
'"
200
...
...
-'!-
... -?-
·n
Every odd number is one less than an even number. 1,3,5.7,9, ... are each one
less than a term in the even sequence 2,4,6,8, 10, .... Therefore, the value of
each term of the sequence of odd nwnbers is one less than twice the number of the
term. The value of the 20th term is 2(20) - 1 or 39. The value of the 200th term is
2(200) - I or 399. In general, the value of the nth term is 2n - 1.
EXERCISE SET A
Find the nth term in ea.h sequence.
1-;*
n
2..
1
2
3
4
5
6
...
5
6
7
8
9
10
... -?-
n
3.
-?-
To find the rule or formula that describes the sequence above, notice that the value of
each tennis three less than the number of the term, Therefore, the 20th term of the
sequence will be 20 .; 3 or 17. The value of the 200th tenn will be 197. In general,
the value of the nth term is n - 3.
50 I Chapter 1
(
4
ExBmpleC'"
Example A
2
3
SometimeS the patterns are created by a combination of addition 01' subtraction with
multiplication. The pattern beloW is the familiar sequence of odd numbers.
n
In previous lessons you found the next term of many sequences, What if you needed
to know the value of the 20th, 200th, or 2000th term of a number sequence but you
didn't know the rule? You certainly don't want to calculate all the values in between
just to get one answer. What you need is a formula or rule that describes the
sequence of values. Let's look: at a few examples to See how you find these rules.
The simplest of the formulas involve simple addition, subtraction, or multiplication
by an integer.
1
2
To find the rule or formula that descnl:Jes the sequence above, notice that the value of
each term is twice the value of the nwnber of the term. The value of the 20th term of
the sequence is 2(20) or 40. The value of the 200tb term is 400. In general, the
value of the nth term is 211. This pattern is the sequence of even numbers.
Each number in the top row represents a term or position number in the sequence.
The number beneath each term number is the value of that term of the sequence. The
n and the n + 2 tell us the rule that connects the top row with the bottom row. This
particular rule states that to find the value of any particular term in this sequence, you
simply add two to the term number. Therefore, if you wish to know the value of the
twentieth term oftbis sequence, you add 20 + 2 to get 22. The value of the twentieth
term of this sequence is 22.
Term
Value
1
1
2
3
4
5
6
...
0
1
2
3
4
5
... -?-
1
2
3
4
5
6
4
7
10
13
16
19
...
...
n
4.
4
1
2
3
5
6
4
8
12 16 20
24
... n
... -?-
n
-?-
Inductive Reasoning I 51
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Lesson 1.4
Finding the nth Term
ExampleB
(jot! trUU4 tfu inUlJers, aff we i
'UI01t
of mono
Tenn
Value
- £eopoM ~rone
The pattern of numbers 3, 4, 5, 6, 7. 8, 9, ... is an example of a sequence of
numbers. The first term of the sequence has a vaI"ueof three, the second term has a
value of four, the third term has a value of five, and so on. Let's show this as a
table.
Term
1
2
3
4
5
6
..
20
Value
3
4
5
6
7
8
...
22
,
...
...
200
...
202
... n+2
3
4
5
6
-2
-1
0
1
2
3
...
...
20
-?-
...
...
200
-1-
...
...
5
6 ....
20
...
200
...
n
2
4
6
8
10
12
...
-?-
...
-1-
...
-7-
Term
1
Value
1
2
4
3
3
5
7
5
6
9
11
...
...
20
-?-
'"
200
...
...
-'!-
... -?-
·n
Every odd number is one less than an even number. 1,3,5.7,9, ... are each one
less than a term in the even sequence 2,4,6,8, 10, .... Therefore, the value of
each term of the sequence of odd nwnbers is one less than twice the number of the
term. The value of the 20th term is 2(20) - 1 or 39. The value of the 200th term is
2(200) - I or 399. In general, the value of the nth term is 2n - 1.
EXERCISE SET A
Find the nth term in ea.h sequence.
1-;*
n
2..
1
2
3
4
5
6
...
5
6
7
8
9
10
... -?-
n
3.
-?-
To find the rule or formula that describes the sequence above, notice that the value of
each tennis three less than the number of the term, Therefore, the 20th term of the
sequence will be 20 .; 3 or 17. The value of the 200th tenn will be 197. In general,
the value of the nth term is n - 3.
50 I Chapter 1
(
4
ExBmpleC'"
Example A
2
3
SometimeS the patterns are created by a combination of addition 01' subtraction with
multiplication. The pattern beloW is the familiar sequence of odd numbers.
n
In previous lessons you found the next term of many sequences, What if you needed
to know the value of the 20th, 200th, or 2000th term of a number sequence but you
didn't know the rule? You certainly don't want to calculate all the values in between
just to get one answer. What you need is a formula or rule that describes the
sequence of values. Let's look: at a few examples to See how you find these rules.
The simplest of the formulas involve simple addition, subtraction, or multiplication
by an integer.
1
2
To find the rule or formula that descnl:Jes the sequence above, notice that the value of
each term is twice the value of the nwnber of the term. The value of the 20th term of
the sequence is 2(20) or 40. The value of the 200tb term is 400. In general, the
value of the nth term is 211. This pattern is the sequence of even numbers.
Each number in the top row represents a term or position number in the sequence.
The number beneath each term number is the value of that term of the sequence. The
n and the n + 2 tell us the rule that connects the top row with the bottom row. This
particular rule states that to find the value of any particular term in this sequence, you
simply add two to the term number. Therefore, if you wish to know the value of the
twentieth term oftbis sequence, you add 20 + 2 to get 22. The value of the twentieth
term of this sequence is 22.
Term
Value
1
1
2
3
4
5
6
...
0
1
2
3
4
5
... -?-
1
2
3
4
5
6
4
7
10
13
16
19
...
...
n
4.
4
1
2
3
5
6
4
8
12 16 20
24
... n
... -?-
n
-?-
Inductive Reasoning I 51
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Lesson 1.5
('\
The triangular number sequence doesn't always begin with a one. What would be the
rule for the triangular number sequence: 6, 10, 15, 21,28, ... '1 Let's look.
Triangular Numbers
~.'
.'
ExampleS
•
••
• •••3 •••
6
•
••
•
•••••
10
15
••• •••••
••••
••••
•
••
•••
••••
•••••
••••••
21
The sequence above appears in many different geometry problems. The sequence is
called the triangular numbers. The ancient Greeks were probably the first to work
with the triangular number sequence. Let's look at how you can find the rule for this
sequence of triangular numbers.
Example A
In this pattern it is easy to get the next term in the sequence, but not so easy to find
the rule that describes the term. You can try factoring 1,3,6,10,15,21, hoping for
a pattern to emerge that relates the terms to the factors.
Value
1
1
Factored
1- 1
Term
·2
3
4
5
6
3
6
10
15
21
1- 3
1• 6
2·3
... 200 ... n
... -7- ... -7- ... -?-
... 20
2
3
4
5
6
Value
1
3·
6
10
15
21
Doubled
2
6
12
20
30
42
2
3
4
5
6
Value
6
10
15
21
28
36
Doubled
12
20
30
42
56
72
ED
... 20 ... 200 ... n
... -?- ... -7- ... -7-
I· 12 1- 20 1- 30 1· 42
2·6 2-10 2·15 2·21
3-10 3-14
90
Since the nth term of the doubled sequence is n(n + 1), then half of that gives the nth
term of the triangular number sequence. The nth triangular number is
54 I
Chapter
1
€I)
EXERCISE SET A
n('!. + 1)
(
2.
1
2
3
4
5
6
0
1
3
6
10
15
...
...
...
n
n
1
2
3
4
5
6
-7-
0
2
5
9
14
20
1
2
3
4
5
6
.. ,
2
5
9
14 20
27
... -7-
1
2
3
6
... n
... -7-
,
.. -'1-
4.
3.
1
2
3
4
3
6
10
15 21
5
... n
28 ... -76
n
6.*-
5.
-1
e·ee
1- 6
20
sa
1
Factored
... 200 ... n
... -1- ... -7- .., -1-
.. ,
Factored 1 - 12 1 - 20 1 - 30 1- 42 1- 42 1- 72
2 - 6 2 - 10 2· 15 2 - 21 2 - 28 2 - 36
3 - 10 3 - 14 4 - 14 3·24
.
~cr:D4'18
6·12
1.*
But this doesn't seem to be too helpful. There is a pattern, but it appears to alternate
(one pattern with odd terms end another with even terms). H factoring doesn't help,
it may be because the formula involves division as well as factoring. If you suspect
that the rule involves division, try doubling or tripling the values, and then factor
these numbers. If the doubled values give you a set of numbers that can be more
readily put into a formula, then this formula must be divided in half to get the formula
for your original sequence. This works with triangular numbers.
1
1
Find the nth term in each sequence.
1· 10 1· 15 1- 21
2·5
3-5 3·7
Term
To find the rule that describes the triangular number sequence below, you would
probably first look at the factors, but you would again have no luck. If you suspect
that the rule involves division, you should try doubling or tripling the values, and
then factor these numbers. The doubled values will again give you a set of numbers
that can be more readily put into a formula. This formula must be divided in half to
get the formula for your originaIsequence.
Tenn
2
-1
3
0
4
2
5
5
6
9
... n
... -7-
2
3
2
4
4
20
5
'3 10
14
7••
~.~.~
,
In this lesson you have learned some new techniques for fmding formulas for
patterns involving division. In all the exercises we have restricted the rules to those
that can be factored because most of the patterns you will encounter while making
geometric discoveries in later chapters will be of this type. There are other techniques
that can be used to find formulas for sequences that follow other rules.
Inductive Reasoning I 55