Activity
1. Clare takes a piece of paper with length 8 inches and width 10 inches and cuts it in half. Then she cuts it in half again, and again . . .
a. Instead of writing a recursive definition, Clare writes , where is the area, in square inches, of the paper after cuts. Explain where the different terms in her expression came from.
Compare your answers:
Each cut halves the area, and there are cuts. The original area is 80, so with multiplications of is the same as saying .
b. Approximately what is the area of the paper after 10 cuts?
Compare your answers:
Approximately 0.078 square inches (exactly inches)
2. Kiran takes a piece of paper with length 8 inches and width 10 inches and cuts away 1 inch of the width. Then he does it again, and again . . .
a. Complete the table for the area of Kiran’s paper , in square inches, after cuts.
0 |
80 |
1 |
|
2 |
|
3 |
|
4 |
|
5 |
Compare your answers:
When is 1, is 72. When is 3, is 56. When is 4, is 48. When is 5, is 40.
b. Kiran says the area after 6 cuts, in square inches, is . Explain where the different terms in his expression came from.
Compare your answers:
Each cut removes 8 square inches, so 6 cuts removes 48 square inches. The original area was 80 square inches, so the remaining area, in square inches, is .
c. Write a definition for that is not recursive.
Compare your answers:
for integers or equivalent.
3. Which is larger, or ?
Compare your answer:
is larger.
4. Is arithmetic or geometric?
Compare your answer:
is arithmetic.
5. Is arithmetic or geometric?
Compare your answer:
is geometric.
A recursive sequence is a sequence in which terms are defined using one or more of the previous terms. If you know the nth term of an arithmetic or geometric sequence and the common difference or factor, you can find the (n+1)th term by using the recursive formula.
Different definitions can often create the same sequence. For arithmetic and geometric sequences, there are general rules, called explicit rules, that can be followed to help find any term in the sequences.
These are called the nth term or the general term of the sequence.
Arithmetic Sequence Formulas
Recursive Formula
, = first term
where is the first term, is the term you want, and is the common difference.
Explicit General Formula
Where is the first term, is the term you want, and is the common difference.
Self Check
Additional Resources
GENERAL TERM (NTH TERM) OF AN ARITHMETIC SEQUENCE
The general term of an arithmetic sequence with first term and common difference is: .
Example 1
Write the explicit formula for the sequence given by the terms 18, 21, 24, 27, ...
Solution
Step 1- Write the general formula.
Step 2 - Substitute values for the first term, common difference/ratio, term number.
,
So, the explicit formula for the sequence 18, 21, 24, 27… is .
Now any term can more easily be determined - even the 1000th term!
Example 2
Find the 15th term of a sequence where the first term is 3 and the common difference is 6.
Step 1 - Write the general formula.
Step 2 - Substitute values for the first term, common difference/ratio, term number.
,
Step 3 - Simplify the expression.
If the explicit formula for this question was needed, we would not have substituted and the th term formula would have been . This formula is also equivalent to
Try it
Try It: The nth Term of Arithmetic Sequences
Find the formula for the th term of a sequence where the first term is 7 and the common difference is 9.
Compare your answer:
Here is how to find the 27th term of the sequence:
Step 1 - Write the general formula.
Step 2 - Substitute values for the first term, common difference/ratio, term number.
,