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Warm UP!

Warm UP!. Indentify the following as Arithmetic, Geometric, or neither: 2, 5, 8, 11, … 2, 6, 24, … 5, 10, 20, 40, … 2. Demonstrate you know the difference in an arithmetic series and sequence by writing one of each. Both should have a first term of 2 and common difference of 3.

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Warm UP!

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  1. Warm UP! Indentify the following as Arithmetic, Geometric, or neither: 2, 5, 8, 11, … 2, 6, 24, … 5, 10, 20, 40, … 2. Demonstrate you know the difference in an arithmetic series and sequence by writing one of each. Both should have a first term of 2 and common difference of 3. 3. Find the 50th term of the sequence an = 2n – 6

  2. Notes The sum of the terms from 1 to n of an arithmetic sequence can be found using this formula: Partial Sum of an Arithmetic Series n=number of terms

  3. Example: Find the Sum • Find the sum: 1+3+5+7+9+11+13+15+17+19

  4. Example: Find the Sum • Find the sum of the integers from 1 to 100.

  5. Summation Notation • AKA Sigma Notation • i = Index of Summation (lower limit) • n = upper limit

  6. Example 6: Sigma Notation

  7. Discussion Find the 78th partial sum:

  8. Practice Evaluate the sum:

  9. Example: Find the Partial Sum • Find the 150th partial sum of the AS. • 5,16,27,38,49,…

  10. Example: An A.S. has a1 = 19 and a2 = 32 • Find the common difference • Find a300 • Find S300 • Find n if an = 7377 • Find n if Sn = 1086 13 3,906 588,750 567 12

  11. Example: Find the Sum • Find the sum: 81+77+73+…+5 860

  12. Example: Applications • An auditorium has 20 rows of seats. There are 20 seats in the 1st row, 21 seats in the second row, 22 seats in the 3rd row, and so on. How many seats are there in all 20 rows?

  13. Example: Applications • A small business sells $10,000 worth of products during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 19 years. Assuming that this goal is met, find the total sales during the first 20 years this business is in operation.

  14. Example: Applications • Shelbi starts an exercise program. At the first workout she does 5 push ups. The next workout she does 8 push ups. She decides to let the number of push ups in each workout be a term in a sequence. Find the number of pushups she does on the tenth workout. Also, find the total number of pushups she has done after 10 workouts. 32 185

  15. sum of a finite geometric series

  16. Ex. 6 Find the sum: = 5.714

  17. Ex. 6 Find the sum: = 98,301

  18. Find the sum of the first 60 terms of the sequence: an = -3( 2 )n-1 = -49,149

  19. Think about it… The ratio of the sum of two arithmetic series is (5n+4):(9n+6). Find the ratio of the 18th partial sums.

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