Group 1: - Long Branch Public Schools
Arithmetic Sequences
Group 1: Sequence: 1, 5, 9, 13, 17, …
1) State the pattern in words:
The first term is _____. Then _______(add/subtract) _________ to the previous term.
2) State the recursive formula:
a1 = _____
an = an-1 + __________
3) What is the initial term? 4) What is the common difference?
5) What is the 10th term? 6) If n = 12, what is an? 7) What is an such that n = 15?
8) Complete the table: 9) Plot these as ordered pairs on a coordinate plane.
|n |an |
|1 | |
|2 | |
| |9 |
| |13 |
|5 | |
|6 | |
|7 | |
|10 | |
|12 | |
|15 | |
10) Take term a2. How many times did you use the common difference, starting from a1?
11) Take term a5. How many times did you use the common difference, starting from a1?
12) Take term a10. How many times did you use the common difference, starting from a1?
13) Take term an. How many times did you use the common difference, starting from a1?
14) Take a2 = 5.
a2 = 1 + ____ = 1 + _____ * _____ =5
15) Take a3 = 9
a3 = 1 + _____ = 1 + ____*_____ = 9
16) Take a10 = ______
a10 = 1 + ______ = 1 + _____ *______ = ____________
17) Take an
an = 1 + _____*______ = _______________
18) So, in summary,
Our recursive formula was:
_______________________
_______________________
A formula that we can use instead, to find an is
an = ________________________
Arithmetic Sequences
Group 2: Sequence: -3, 2, 7, 12, 17…
1) State the pattern in words:
The first term is _____. Then _______(add/subtract) _________ to the previous term.
2) State the recursive formula:
a1 = _____
an = an-1 + __________
3) What is the initial term? 4) What is the common difference?
5) What is the 10th term? 6) If n = 12, what is an? 7) What is an such that n = 15?
8) Complete the table: 9) Plot these as ordered pairs on a coordinate plane.
|n |an |
|1 | |
|2 | |
| |7 |
| |12 |
|5 | |
|6 | |
|7 | |
|10 | |
|12 | |
|15 | |
10) Take term a2. How many times did you use the common difference, starting from a1?
11) Take term a5. How many times did you use the common difference, starting from a1?
12) Take term a10. How many times did you use the common difference, starting from a1?
13) Take term an. How many times did you use the common difference, starting from a1?
14) Take a2 = 2.
a2 = -3 + ____ = -3 + _____ * _____ =2
15) Take a3 = 7
a3 = -3 + _____ = -3 + ____*_____ = 7
16) Take a10 = ______
a10 = -3 + ______ = -3 + _____ *______ = __________________
17) Take an
an = -3 + _____*______ = ____________________
18) So, in summary,
Our recursive formula was:
_______________________
_______________________
A formula that we can use instead, to find an is
an = ________________________
Arithmetic Sequences
Group 3: Sequence: 9, 2, -5, -12, -19…
1) State the pattern in words:
The first term is _____. Then _______(add/subtract) _________ to the previous term.
2) State the recursive formula:
a1 = _____
an = an-1 + __________
3) What is the initial term? 4) What is the common difference?
5) What is the 10th term? 6) If n = 12, what is an? 7) What is an such that n = 15?
8) Complete the table: 9) Plot these as ordered pairs on a coordinate plane.
|n |an |
|1 | |
|2 | |
| |-5 |
| |-12 |
|5 | |
|6 | |
|7 | |
|10 | |
|12 | |
|15 | |
10) Take term a2 How many times did you use the common difference, starting from a1?
11) Take term a5. How many times did you use the common difference, starting from a1?
12) Take term a10. How many times did you use the common difference, starting from a1?
13) Take term an. How many times did you use the common difference, starting from a1?
14) Take a2 = 2.
a2 = 9 + ____ = 9 + _____ * _____ = 2
15) Take a3 = -5
a3 = 9 + _____ = 9 + ____*_____ = -5
16) Take a10 = ______
a10 = 9 + ______ = 9 + _____ *______ = __________________
17) Take an
an = 9 + _____*______ = __________________
18) So, in summary,
Our recursive formula was:
_______________________
_______________________
A formula that we can use instead, to find an is
an = ________________________
Arithmetic Sequences
Part 2: Group discussion
1) Complete the chart below with your fellow group members’ problems
|Group number |Sequence |Recursive |Explicit |
|1 | | | |
|2 | | | |
|3 | | | |
2) For each of the groups, use the explicit formula to find a) a15 and b) decide if an = 80 is part of each sequence.
3) What do you notice about all of the graphs of these arithmetic sequences? How do the initial term and the common difference appear in each of the graphs?
4) In general, given a recursive formula
a1 = a1
an = an-1 + d
Write an explicit formula to find an
an = _____________________________
Try these examples: Write a recursive formula and explicit formula for each sequence. Simplify each explicit formula. Then use your formula to find the 100th term of the sequence. Then determine if the number 30 is part of the sequence or not.
1) 7, 10, 13, 16, …
2) 0 , 3 , 6 , 9 , ...
3) 0.9 , 0.5 , 0.1 , 0.3 , ...
4) 3.2 , 3.5 , 3.8 , 4.1 , ..
Use what you know to solve these word problem.
1) Suppose you participate in a BikeaThon for charity. The charity starts with $1100 in donations. Each participant must raise $35 in pledges.
a. How much money does the charity have with 0 participants? With 1 participant?
b. Write a recursive formula for this sequence. What does n represent in this situation? What does an represent?
c. Write an explicit formula for this sequence.
d. What is the 75th term in the sequence? What does it represent?
2) Elliot borrowed $370 from his parents. He will pay them back at the rate of $60 per month. Write an explicit formula to determine how long will it take for him to pay his parents back?
Write a recursive formula for each of these explicit formulas.
1) Given an = 4 + 3(n -1), write the recursive formula
2) Given an = 7 – 2(n – 1), write the recursive formula.
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