Geometric Sequences 1©Relevantmathematics.com Geometric Sequences We don’t really think about it, but we rely on the basketball’s ability to bounce back to a specific height. If basketballs bounced to radically different heights, we couldn’t do things like this:

Kaleb’s Work 2©Relevantmathematics.com Geometric Sequences

Sandra’sWork 3©Relevantmathematics.com Geometric Sequences

Try one Consider the following sequence: 4, 8, 16, 32, … a. Write a recursive rule for this sequence. b. Write an explicit rule for this sequence. c. Use one of your rules to find the 6 th term in the sequence. a.Let n = number of term Let f(n) = value of term n f(1) = 10 f(n) = 3 ∙ f(n) for integers n ≥ 2 b.f(n+1) = 10 ∙ 3 (n) for integers n ≥ 1 c. f(6) = 10 ∙ 3 (6) f(6) = 10 ∙ 729 f(6) = ©Relevantmathematics.com Geometric Sequences

Your cell phone is covered in bacteria. The same bacteria from your fingers and you face also live on your cell phone. A culture of bacteria swabbed from a cell phone has 150 of a particular bacteria on it. That particular bacteria doubles every day. a. Write a recursive rule to represent this situation. b. Write an explicit rule to represent this situation. c. Use both of your rules to determine the total amount of bacteria after 8 days. 5©Relevantmathematics.com Geometric Sequences

Geometric Sequence: is a sequence of numbers such that each term is given by a common multiple, r, of the previous term. f(n + 1) = r · f(n) where n is the number of the term. Recursive rule for a geometric sequence: Let n be the number of the term. Let f(n) be the value of term n. Let p be the value of the first term f(1) = p f(n + 1) = r · f(n) for integers n ≥ 2 Explicit rule for a geometric sequence: f(n) = (r) n-1 · f(1), for integers n ≥ 1 or f(n + 1) = (r) n · f(1), for integers n ≥ 1 or a closed formula: f(n) = f(0) · r n, for integers n ≥ 1 ©Relevantmathematics.com6 Geometric Sequences

Recall.. Arithmetic Sequences Arithmetic Sequence: a sequence of numbers such that each term is given by a common difference, d, of the previous term. f(n + 1) = f(n) + d where n is the number of the term. Recursive rule for an arithmetic sequence:. Let n be the number of the term. Let f(n) be the value of term n. Let p be the value of the first term f(1) = p f(n + 1) = f(n) + d for integers n ≥ 2 or f(1) = p f(n) = f(n - 1) + d for integers n ≥ 2 Explicit Rule for an arithmetic sequence: f(n + 1) = d(n) + f(1), for integers n ≥ 1 or f(n) = d(n-1) + f(1), for integers n ≥ 1 or a closed formula: f(n) = dn + f(0), for integers n ≥ 1 ©Relevantmathematics.com7 Geometric Sequences

In your groups, take turns to matching a situation card to one of the function cards. For each recursive function, define n and f(n). For each blank card, fill in the appropriate information to create a matched set. If you place a card, explain why that situation matches the function. Everyone in your group should agree on, and be able to explain, your choice. Then, take turns matching your explicit function to your matched pair of recursive functions and situations. Wherever there are blank cards, create an explicit or recursive function to make a matching set. When everyone agrees, glue the cards across as a single row in the respective columns on your paper. ©Relevantmathematics.com8 Geometric Sequences