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Arithmetic Sequences

Arithmetic Sequences. Chapter 3-4. Vocabulary. A set of numbers in a specific order is a SEQUENCE. The numbers in the sequence are called TERMS If the difference between terms is constant (the same), then it is called an ARITHMETIC SEQUENCE. Arithmetic Sequences.

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Arithmetic Sequences

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  1. Arithmetic Sequences Chapter 3-4

  2. Vocabulary • A set of numbers in a specific order is a SEQUENCE. • The numbers in the sequence are called TERMS • If the difference between terms is constant (the same), then it is called an ARITHMETIC SEQUENCE.

  3. Arithmetic Sequences • In order to see if a sequence is an arithmetic sequence, find the difference between each term to see if it is the same. If it is the same than it is an arithmetic sequence. Ex1: Is this an arithmetic sequence? Answer: No this is not an arithmetic sequence because the difference between the terms is not the same (constant)

  4. Arithmetic Sequences • In order to see if a sequence is an arithmetic sequence, find the difference between each term to see if it is the same. Ex1: Is this an arithmetic sequence? Answer:Yes, this is an arithmetic sequence because the difference between terms is constant.

  5. Arithmetic Sequences • The difference between terms in an arithmetic sequence is called the COMMON DIFFERENCE. • What is the common difference for the following sequence? Common Difference: +2

  6. Writing Arithmetic Sequences • In words: • You can use the common difference of an arithmetic sequence to find the next term by adding it to the previous term • In symbols: (d is the common difference, a1 is the first term, a2 is the second term, a3 is the third term and so on.) • a1, a1+d, a2+d, …, Ex: If the first term in an arithmetic sequence is 8 and the common difference is 4 find the next 3 terms. Answer: 8, 12, 16, 20 +4 +4 +4

  7. Writing Arithmetic Sequences • Example: The arithmetic sequence –8, –11, –14, –17, … represents the daily low temperature in ºF. Find the next three terms. First find the common difference by subtracting successive terms. The common difference is –3. • Second add -3 to the last term to get the next three terms. (remember adding -3 is the same as subtracting 3)

  8. Try some on your own • The arithmetic sequence 58, 63, 68, 73, … represents the daily high temperature in ºF. Find the next three terms. • The arithmetic sequence 74, 67, 60, 53, … represents the amount of money tiffany owes her mother at the end of each week, find the next terms. • Find the next three terms for the following arithmetic sequence 9.5, 11.0, 12.5, 14.0,…

  9. Exit-Slip • Describe what an arithmetic sequence is and how you find it. • What did you learn today? What are you still having trouble with?

  10. Nth Term in a sequence Each term in a sequence can be expressed in terms of the common difference d, and the first term in the sequence a1

  11. Nth term in a sequence • The previous table leads to the equation to find the nth term in a sequence.

  12. Example 1 • The arithmetic sequence 1,10,19, 28, … is used to represent the total number of dollars Erin has in her bank account after her weekly allowance was added. Write an equation for the nth term in the sequence. • In order to write the equation we need a1 and d • What is a1 in this sequence? What is d in this sequence? d= +9

  13. Still example 1 • Use what we know to write the equation: • We know a1=1, d=9 and • we know the formula an= a1 +(n-1)(d) • Plug in everything you know an = a1 + (n –1)d Formula for the nth term an = 1 + (n –1)(9) a1 = 1, d = 9 an = 1 + 9n – 9 Distributive Property an = 9n – 8 Simplify.

  14. Using the equation you got in example 1, Find the 12th term in the sequence Wherever you see an n replace n with 12 in the equation. an = 9n – 8 Equation for the nth term a12= 9(12) – 8 Replace n with 12. a12= 100 Simplify.

  15. Using the same example graph the first five terms.

  16. Try one on your own • The arithmetic sequence 2, 7, 12, 17, 22, … represents the total number of pencils Claire has in her collection after she goes to her school store each week. • Write an equation for the nth term in the sequence. • Find the 12th term in the sequence using the equation you found. • Graph the first five terms (n, an)

  17. Exit-Slip • How do you think arithmetic sequences relates to linear equations? What is similar about them? What is different? • What did you learn today? What do you still need help with?

  18. Chapter 3-5 Proportional and Non-Proportional

  19. Relationships • Proportional Relationship- • A relationship that can be expressed as y=kx • The graph always passes through the point (0,0) • Non-proportional relationship • A relationship in which a constant has to be added or subtracted • looks like y=kx+c

  20. How do you know if a relationship is proportional or non-proportional? • In a proportional relationship is always the same or the graph passes through the origin (0,0). • In a non-proportional relationship is not always the same or does not pass through the origin.

  21. Try some on your own (5 mins) • Determine if the following sequences are arithmetic sequences, if they are find the common difference. • 2, 4, 8, 10, 12 • -26, -22, -18, -14

  22. Let’s play proportional or non-proportional. Answer: Yes it is a proportional relationship because all the hours/miles are the same.

  23. Let’s play proportional or non-proportional. Answer: Yes it is a proportional relationship because all the hours/miles are the same.

  24. Let’s play proportional or non-proportional. Answer: No, not proportional because the graph does not pass through the origin

  25. Let’s play proportional or non-proportional. Answer: No not a proportional relationship because

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