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Chapter 9.2: Arithmetic Sequences and Partial Sums

Chapter 9.2: Arithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition: if the difference between consecutive terms are the same then the sequence is arithmetic. Sequence a 1 , a 2 , a 3 , a 4 , . . ., a n is arithmetic if a 2 – a 1 = a 3 – a 2 = a 4 – a 3 = d

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Chapter 9.2: Arithmetic Sequences and Partial Sums

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  1. Chapter 9.2: Arithmetic Sequences and Partial Sums

  2. Arithmetic Sequences • Definition: if the difference between consecutive terms are the same then the sequence is arithmetic. • Sequence a1, a2, a3, a4, . . ., an is arithmetic if • a2 – a1 = a3 – a2 = a4 – a3 = d • The number d is the common difference of the arithmetic sequence.

  3. Examples of Arithmetic Sequences • Find the common difference and the formula for the sequence: • 7, 11, 15, 19 • Common difference is 4 and our formula is 4n+3 • 2, -3, -8, -13 • Common difference is -5 and our formula is 7-5n • 1, 5/4, 3/2, 7/4 • Common difference is ¼ and the formula is n+3/4

  4. Finding the nth term of an Arithmetic Sequence • The nth term of an arithmetic sequence has the form • an = dn +c • where d is the common difference between consecutive terms of the sequence and c = a1 – d • Alternative form • an = a1 + (n + 1)d

  5. Find the nth term of the Arithmetic Sequence • Find a formula for the nth term of the arithmetic sequence whose common difference is 3 and whose first term is 2. • You try: a1 = 100, d = -8

  6. Writing the Terms of an Arithmetic Sequence • The fourth term of an arithmetic sequence is 20, and the 13th term is 65. Write the first 11 terms of this sequence. • You try: Write the first five terms of the arithmetic sequence: a8 = 26, a12 = 42

  7. Using a Recursion Formula • Find the ninth term of the arithmetic sequence that begins with 2 and 9. • You try: a1 = 5, a2 = 11, a10 = ____

  8. The Sum of a Finite Arithmetic Sequence • The sum of a finite arithmetic sequence with n terms is • Find the sum: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15+ 17 + 19 • You Try: a1 = 100, a25 = 220, n = 25

  9. Finding the Sum of a Finite Arithmetic Sequence • Find the sum of the integers from 1 to 100 • Find the sum of the integers from 1 to N • You Try: Find the sum of the first 100 positive odd integers

  10. Finding a Partial Sum of an Arithmetic Sequence • Find the 150th partial sum of the arithmetic sequence 5, 16, 27, 38, 49 • You Try:

  11. In a golf tournament, the 16 golfers with the lowest scores win cash prizes. First place receives a cash prize of $1000, second place receives $950, third place receives $900, and so on. What is the total amount of prize money?

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