2.3 Finding the nth Term
__________________________- A rule that, when applied to one set of numbers (the input),
generates another set of numbers (the output).
____________________________- A rule that, when applied to consecutive whole numbers,
generates a sequence with a constant difference between consecutive terms.
Investigation: Finding the Rule
Step 1 Complete each table. Find the differences between consecutive values.
a)
n
n-5
1
-4
2
-3
3
-2
4
5
6
7
8
b)
n
4n-3
1
1
2
5
3
9
4
5
6
7
8
c)
n
-2n+5
d)
n
3n-2
e)
n
-5n+7
1
3
2
1
1
1
3
-1
2
4
1
2
4
3
7
2
-3
5
4
6
5
3
-8
4
7
6
5
8
7
6
8
7
8
Step 2 Did you spot the pattern? If a sequence has a constant difference of 4, then the number in
front of n (the coefficient of n) is _____. In general, if the difference between the values of
consecutive terms of a sequence is always the same, say m (a constant), then the coefficient of
n in the formula is _______.
Lets look at the sequence below:
Term
1
2
3
Value
20
27
34
4
41
5
48
6
55
7
62
…
…
n
The constant difference is 7, so you know part of the rule is 7n. How do you find the rest of the rule?
Geometry Lesson 2.3: Finding the nth Term
Page 1
Step 3 The first term (n = 1) of the sequence is 20, but if you apply the part of the rule you have so
far using n = 1, you get 7n = 7 ( 1 ) = 7, not 20. So how should you fix the rule? How can
you get from 7 to 20? What is the rule for this sequence?
Step 4 Check your rule by trying the rule with other terms in the sequence.
Example 1: Find the function rule for each sequence.
a) 7, 2, -3, -8, -13, -18, …
term
value
2
3
4
5
6
…
…
n
1
2
3
4
5
6
…
…
n
1
2
3
4
5
6
…
…
n
1
b) 10, 14, 18, 22, 26, …
term
value
c) -1, 2, 5, 8, 11, 14
term
value
Geometry Lesson 2.3: Finding the nth Term
Page 2
Example 2: If you place 200 points on a line, into how many non-overlapping rays and segments
does it divide the line?
Points dividing the line
Non-overlapping rays
Non-overlapping segments
Total
1
2
3
4
5
6
…
…
…
…
n
…
…
…
…
200
Example 3: If the pattern of T-shapes continues, how many squares will be in the 100th shape?
T-shape
1 2 3 4 … n … 100
Number of squares
pp. 109 – 110 => 1 – 8; 10 - 16
Geometry Lesson 2.3: Finding the nth Term
Page 3