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Understanding Recursive and Explicit Functions in Sequences

This review module explores recursive and explicit forms of sequences through practical examples. Key calculations include finding common differences and ratios, as well as analyzing sequences related to savings and expenses. Specifically, we derive explicit functions from given recursive definitions, graph sequences, and solve real-life problems such as savings over time. The module emphasizes the arithmetic nature of savings and the associated calculations for a clearer comprehension of sequences in financial contexts, including strategies for saving for major purchases.

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Understanding Recursive and Explicit Functions in Sequences

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  1. Module 3 Test Review

  2. Using the sequence 9, 14, 19, 24… • Write the Recursive Form: • Answer f(n) = f(n-1) + 5 • Write the Explicit Form: • Answer f(n) = 5n + 4 • Graph the Sequence • Answer

  3. Given the Recursive Function f(n) = f(n-1) + 11 and f(1) = 3 • Write the Explicit Function • Answer: f(n) = 11n - 8

  4. Given the Explicit Function f(n)= · -2 • Write the Recursive Form: • Answer f(n) = f(n-1) · 3 and f(1) = -2

  5. Given • Find the Common Difference and the Arithmetic Means • Answer: Common Difference = 7

  6. Given • Find the Common Ratio and the Geometric Means • Answer: Common Ratio = 3

  7. Von is starting to save for a new car. He decides that every month he is going to put the same amount of money into a savings account each month. • What type of sequence is represented by his total savings? • Answer: Arithmetic. He is adding the same amount each month. • After week 2 he had $300 and after week 7 he had $1050. What is the Common Difference and his values of Arithmetic Means? • Answer: Common Difference = $150 • How many months will it take Von to save $3000? • Answer: 3000 = 150n • n = 20 months

  8. Von bought a car for $5000 and agreed to pay the owner 15% each month. • How much did he still owe after 4 months? • Answer: f(n) = · 4250 • f(4) = $2610.03

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