1. Patterns In Sequence
By Miss Seymour

2. Objectives:
To recognize, describe, and extend patterns in sequences

3. Vocabulary
Triangular number: a number that can be represented by a triangular array such as 1,2,6,10 and so forth
Sequence: an ordered set of numbers
Term: each number in a sequence is called a term.

4. Why learn about patterns in sequences?
A pattern can be used to determine the increase or decrease in value a collectible item might have after a period of time.

5. Triangular Arrays Question:
How do the arrays on the right change from one to the next?
Answer:
A new row of circles is placed under the previous triangle. Each new row has one more circle to keep the triangle shape.

6. Reasoning How are triangular numbers like square numbers?
A square number is a number that can be represented by a square array. 16 is a square number. What are other square numbers?
4,9,25,36,49,etc

7. How to find a pattern Question:
If you know the terms in the sequence, how would you find a pattern?
Answer:
Choose one term of the sequence. Compare it to the previous term. Then, using the original term, compare it to the next term in the sequences. Use these comparisons to look for a rule for finding the next term.

8. Example
 Identify a pattern in the sequence 1, 5 1/2 , 10, 14 1/2 , 19, and write the rule. Using your rule, find the next three terms in the sequence.
Look for a pattern. Compare each term with the next. A rule is to add 4 1/2  to each term to get the next term.
So, 23 1/2 , 28,  32 1/2 are the next three terms.
• What is the tenth term of this sequence?

9. Think and Discuss
Write a rule to make the following sequence. Then find the next three terms. 8, 32, 128, 512, . . .
Multiply by 4; 2,048; 8,192; 32,768.
Tell whether the sequence 1,200; 240; 48; is increasing or decreasing?
It is decreasing.

10. Find the rule and the sixth term:
5, 20, 35, 50, . . .
Add 15 to each successive term
The sixth term is 80.
0.001, 0.01, 0.1, 1, . . .
Multiply each term by 10 to get the next term
The sixth term is 100.

11. Use the rule to write a sequence
Start with 9; add 3.7
9, 12.7, 16.4, 20.1, . . .
Start with 5; multiply by 6
5, 30, 180, 1,080, . . .

12. Put all your skills together!
Mr. Amano started a coin collection. He started his collection with 29 coins and then set a goal to add a certain number of coins each year. The sequence 29, 41, 53, 65, shows the pattern of the number of coins in his collection. Write the rule for the pattern and find the number of coins in his collection in the sixth year.
add 12; 89 coins

13. The End