ARITHMETIC & GEOMETRIC SEQUENCES

43210 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 (n th ) 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. 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.

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 a n = 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: x n = a + d(n-1) (We use “n-1” because d is not used on the 1 st term.)

ARITHMETIC SEQUENCE For each sequence, if it is arithmetic, find the common difference. 1.-3, -6, -9, -12, … 2.1.1, 2.2, 3.3, 4.4, … 3.41, 32, 23, 14, 5, … 4.1, 2, 4, 8, 16, 32, … 1.d = -3 2.d = d = -9 4.Not an arithmetic sequence.

ARITHMETIC SEQUENCE Write the explicit rule for the sequence 19, 13, 7, 1, -5, … Start with the formula: x n = a + d(n-1) a is the first term = 19 d is the common difference: -6 The rule is: x n = (n-1) Find the 12 th term of this sequence. Substitute 12 in for “n.” x 12 = (12-1) x 12 = (11) x 12 = 19 – 66 x 12 = (12-1) x 12 = -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 x n = 2 n

GEOMETRIC SEQUENCE In general we can write a geometric sequence like this: a, ar, ar 2, ar 3, … a is the first term r is the factor between the terms (called the “common ratio”). The rule is x n = ar (n-1) We use “n-1” because ar 0 is the 1 st term.

GEOMETRIC SEQUENCE For each sequence, if it is geometric, find the common ratio. 1.2, 8, 32, 128, … 2.1, 10, 100, 1000, … 3.1, -1, 1, -1, … 4.20, 16, 12, 8, 4, … 1.r = 4 2.r = r = -1 4.Not a geometric sequence.

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