UNIT 1 - ARITHMETIC & GEOMETRIC SEQUENCES
Task #4 – Counting Dots (Finding Rules for nth Term of a Sequence)
Common Core: HS.F-IF.3, HS.F-BF.1a, 2, HS.F-LE.2
MA40: ALGEBRA 2
Name:
Period:
DERIVE THE EXPLICIT FORMULA FOR THE nth TERM OF AN ARITHMETIC SEQUENCE
1. Finding the formula for any term of an arithmetic sequence is essentially looking for a pattern and
making a generalization that fits any and every term in the sequence. Complete the table using
the drawing the dot problem. Find the explicit formula that describes the n th term.
At one
minute
At the
Beginning
Minutes
Term
At the
beginning
1st
term
At one
minute
2nd
term
At two
minutes
3rd
term
At three
minutes
4th
term
At four
minutes
5th
term
…
…
At fourteen
min.
15th
term
…
…
At n  1
minutes
n th
At two
minutes
At three
minutes
At four
minutes
# of Dots
# of dots written
as a sum.
Rewrite the previous column
by combining like terms.
…
…
…
…
…
…
term
2. Use algebra to clean up this formula (simplify) and then use this explicit formula to find the number
of dots at 40 minutes. Use this explicit formula to find the number of dots at 67 minutes.
3. Using a1 to represent the first term and d to represent the common difference, rewrite the
equation derived in problem #2 by substituting in a1 and d. This new equation is the Explicit
Rule for Finding the nth Term of an Arithmetic Sequence.
DERIVE THE RECURSIVE FORMULA FOR THE nth TERM OF AN ARITHMETIC SEQUENCE
4. Complete the table using the drawing of the dot problem. Find the recursive formula that
describes the n th term.
At one
minute
At the
Beginning
At two
minutes
At three
minutes
At four
minutes
Minutes
At the
Beginning
At one
minute
At two
minutes
At three
minutes
A four
minutes
…
At fourteen
minutes
…
Term
1
2
3
4
5
…
15
…
# of Dots
…
At
n  1 minutes
n th
…
5. Compare the table in problems #1 and #4. Why is the table in problem #1 better suited to find
an explicit formula? Why is the table in problem #4 better suited to find a recursive formula?
6. Using a1 to represent the first term and d to represent the common difference, rewrite the
equation derived in problem #4 by substituting in a1 and d. This new equation is the Recursive
Rule for Finding the nth Term of an Arithmetic Sequence.
7. Use this recursive formula to find the number of dots at 6 minutes. Use this recursive formula to
find the number of dots at 15 minutes.
8. Find the explicit and recursive rules for the nth term of the given sequence. Justify your answer
by showing your work.
99, 93, 87, 81, 75, ...
9. Find the explicit and recursive rules for the nth term of the sequence with the first term a1 and the
common difference d.
a1  0.5, d  0.25
DERIVE THE EXPLICIT FORMULA FOR THE nth TERM OF A GEOMETRIC SEQUENCE
10. Finding the formula for any term of a geometric sequence is essentially looking for a pattern and
making a generalization that fits any and every term in the sequence. Complete the table using
the drawing the dot problem. Find the explicit formula that describes the n th term.
At the beginning
At one minute
At three minutes
At two minutes
At four minutes
Minutes
Term
At the
beginning
1st
term
At one
minute
2nd
term
At two
minutes
3rd
term
At three
minutes
4th
term
At four
minutes
5th
term
…
…
At fourteen
min.
15th
term
…
…
At n  1
minutes
n th
# of Dots
# of dots written
as a product.
Rewrite the previous column
using definition of exponents.
…
…
…
…
…
…
term
11. Use this explicit formula to find the number of dots at 20 minutes. Use this explicit formula to
find the number of dots at 25 minutes.
12. Using a1 to represent the first term and r to represent the common ratio, rewrite the equation
derived in problem #2 by substituting in a1 and r. This new equation is the Explicit Rule for
Finding the nth Term of an Geometric Sequence.
DERIVE THE RECURSIVE FORMULA FOR THE nth TERM OF A GEOMETRIC SEQUENCE
13. Complete the table using the drawing the dot problem. Find the recursive formula that describes
the n th term.
At the beginning
At one minute
At three minutes
At two minutes
At four minutes
Minutes
At the
Beginning
At one
minute
At two
minutes
At three
minutes
A four
minutes
…
At fourteen
minutes
…
Term
1
2
3
4
5
…
15
…
…
# of Dots
At
n  1 minutes
n th
…
14. Describe how you used the table to find the recursive formula.
15. Use this recursive formula to find the number of dots at 6 minutes. Use this recursive formula to
find the number of dots at 15 minutes.
16. Find the explicit and recursive rules for the nth term of the given sequence. Justify your answer
by showing your work.
 3, 6,  12, 24, ...
17. Find the explicit and recursive rules for the nth term of the sequence with the first term a1 and
the common ratio r.
a1  3, r 
1
2