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Understanding Arithmetic and Geometric Sequences

Learn about arithmetic and geometric sequences, including their formulas, common differences, and ratios. Explore recursive formulas and practice solving sequence problems.

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Understanding Arithmetic and Geometric Sequences

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  1. Week 4 Warm Up 09.08.11 1) What is the midpoint? ( 2, 5 ) and ( 14, 1 )

  2. Sequence An ordered list of numbers with common differences or ratios. Ex 1 7, 10, 13, 16, 19, . . .

  3. Ex 1 7, 10, 13, 16, 19, . . . n = 4 tn = 16 tn– 1 = 13 common difference Arithmetic Sequence: find the difference ( d ). 16 - 13 = 3 - tn tn– 1 = 3 = tn tn– 1 + 3 Arithmetic Formula = tn tn– 1 + d

  4. Ex 2 -6, 2, 10, 18, 26, . . . 8 = d = = tn tn tn– 1 tn– 1 + 8 + d Check t4 - 1 t4 = + 8 t3 t4 = + 8 10 t4 = + 8 18 t4 =

  5. 4, 12, 36, 108, 324, . . . Ex 3 n = 2 tn = 12 tn– 1 = 4 common ratio Geometric Sequence: find the ratio ( r ) ( ratio means quotient ). tn = 3 tn-1 12 = 3 4 tn tn = 3 = r Geometric Formula tn-1 tn-1

  6. 2, 8, 32, 128, 512, . . . Ex 4 4 = r tn = 4tn-1 Check t3 = 4(t3-1) t3 = 4(t2) t3 = 4(8) tn = r tn-1 t3 = 32

  7. Recursive Formulas You have to know the previous number to find the next number. = Arithmetic tn tn– 1 + d Geometric tn = r tn-1

  8. 3, 10, 17, 24, 31, . . . Arithmetic d = 7 = tn tn– 1 + d tn = tn-1 + 7

  9. Geometric 3, 12, 48, 192, 768, . . . r = 4 tn = 4tn-1 tn = r tn-1

  10. Do: 1 What is the recursive formula for this sequence? -6, 1, 8, 15, 22, . . . Handouts – 1.2 Recursive Formulas Part 2 Assignment: - HW Part 2 - Recursive Formulas

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