Arithmetic & Geometric Sequences

Focus 7 Learning Goal – (HS. F-BF. A. 1, HS. F-BF. A. 2, HS. F-LE. A Focus 7 Learning Goal – (HS.F-BF.A.1, HS.F-BF.A.2, HS.F-LE.A.2, HS.F-IF.A.3) = Students will build a function (linear and exponential) that models a relationship between two quantities. The primary focus will be on arithmetic and geometric sequences. 4 3 2 1 In addition to level 3.0 and above and beyond what was taught in class,  the student may: · Make connection with other concepts in math · Make connection with other content areas. The student will build a function (linear and exponential) that models a relationship between two quantities. The primary focus will be on arithmetic and geometric sequences. - Linear and exponential functions can be constructed based off a graph, a description of a relationship and an input/output table. - Write explicit rule for a sequence. - Write recursive rule for a sequence. The student will be able to: - Determine if a sequence is arithmetic or geometric. - Use explicit rules to find a specified term (nth) in the sequence.   With help from the teacher, the student has partial success with building a function that models a relationship between two quantities. Even with help, the student has no success understanding building functions to model relationship between two quantities.

Arithmetic Sequence In an Arithmetic Sequence the difference between one term and the next term is a constant. We just add some value each time on to infinity. For example: 1, 4, 7, 10, 13, 16, 19, 22, 25, … This sequence has a difference of 3 between each number. It’s rule is an = 3n – 2.

Arithmetic Sequence In general, we can write an arithmetic sequence like this: a, a + d, a + 2d, a + 3d, … a is the first term. d is the difference between the terms (called the “common difference”) The rule is: xn = a + d(n-1) (We use “n-1” because d is not used on the 1st term.)

Arithmetic Sequence For each sequence, if it is arithmetic, find the common difference. -3, -6, -9, -12, … 1.1, 2.2, 3.3, 4.4, … 41, 32, 23, 14, 5, … 1, 2, 4, 8, 16, 32, … d = -3 d = 1.1 d = -9 Not an arithmetic sequence.

Arithmetic Sequence Write the explicit rule for the sequence 19, 13, 7, 1, -5, … Start with the formula: xn = a + d(n-1) a is the first term = 19 d is the common difference: -6 The rule is: xn = 19 - 6(n-1) Find the 12th term of this sequence. Substitute 12 in for “n.” x12 = 19 - 6(12-1) x12 = 19 - 6(11) x12 = 19 – 66 x12 = -47

Geometric Sequence In a Geometric Sequence each term is found by multiplying the pervious term by a constant. For example: 2, 4, 8, 16, 32, 64, 128, … The sequence has a factor of 2 between each number. It’s rule is xn = 2n

Geometric Sequence In general we can write a geometric sequence like this: a, ar, ar2, ar3, … a is the first term r is the factor between the terms (called the “common ratio”). The rule is xn = ar(n-1) We use “n-1” because ar0 is the 1st term.

Geometric Sequence For each sequence, if it is geometric, find the common ratio. 2, 8, 32, 128, … 1, 10, 100, 1000, … 1, -1, 1, -1, … 20, 16, 12, 8, 4, … r = 4 r = 10 r = -1 Not a geometric sequence.

Geometric Sequence Write the explicit rule for the sequence 3, 6, 12, 24, 48, … Start with the formula: xn = ar(n-1) a is the first term = 3 r is the common ratio: 2 The rule is: xn = (3)(2)(n-1) (Order of operations states that we would take care of exponents before you multiply.) Find the 12th term of this sequence. Substitute 12 in for “n.” x12 = (3)(2)(12-1) x12 = (3)(2)(11) x12 = (3)(2048) x12 = 6,144

Group Activity Each group will receive a set of cards with sequences on them. Separate the cards into two columns: Arithmetic and Geometric. For each Arithmetic Sequence, find the common difference and write an Explicit Formula. For each Geometric Sequence, find the common ratio and write a Explicit Formula.

Explain the difference between an Arithmetic and Geometric Sequence.