# Geometric Sequences Teacher Notes

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Geometric Sequences Teacher Notes
Notes: Have students complete the sequence organizer in the debrief Vocabulary: geometric sequence common ratio 3/31/2011 ©Evergreen Public Schools 2010

Geometric Sequences Recursive Formula

Math Practice 7 Target Practice 7. Look for and make use of structure.

Math Practice 8 Target

Learning Target A-SSE I can see structure in expressions.
2-7 Solving Equations With Algebra Tiles powerpoint Learning Target A-SSE I can see structure in expressions. Identify and explain the initial value and common difference of an arithmetic sequence. Translate between sequences written in explicit form and recursive form. A-SSE.1   Interpret expressions that represent a quantity in terms of its context. 1.a   Interpret parts of an expression, such as terms, factors, and coefficients. A-SSE.3 – Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 

2-7 Solving Equations With Algebra Tiles powerpoint
Learning Target F-IFa I can understand the concept of a function and use function notation. Recursive equations require the value of one term and it’s corresponding term number. Sequences can be defined by using either the next term or the previous term. The domain of sequences is the non-negative integers. F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Learning Target F-BF I can build a function.
2-7 Solving Equations With Algebra Tiles powerpoint Learning Target F-BF I can build a function. Write an explicit equation of an arithmetic or geometric sequence. Write a recursive equation of an arithmetic or geometric sequence. F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. 

Launch From section SQ.1 What are the domains of the L(x) and a(x) sequences? The domain for both are the integers greater than or equal to 1. ©Evergreen Public Schools 2010

Launch From section SQ.2 Rumors What is the domains of the rumors? Plan 1 Integers from 1 to 64 Plan 2 Integers from 1 to 16 Plan 3 Integers from 1 to 12 ©Evergreen Public Schools 2010

Launch What do you think is true about the domain of any sequence? The domain for a sequence is positive integers. ©Evergreen Public Schools 2010

Sequences a) 3, 6, 9, 12, 15, … b) 3, 6, 12, 24, 48, … What do the sequences have in common? How are they different? Write an explicit rule with sequence notation for each. First 2 terms the same Both increasing or growing Sequence b grows faster than sequence a a grows by adding 3 each time b grows by multiplying by 2 each time a) an = 3 + 4(n – 1) or an = 4n – 1 b) an = 3½ (2n) or an = 3 (2n-1) ©Evergreen Public Schools 2010

Arithmetic Sequences L(x) = 2x + 1 and N(x) = 34 – 4x are arithmetic sequences. They have a constant rate of change. What is the rate of change of L(x) and N(x)? First 2 terms the same Both increasing or growing Sequence b grows faster than sequence a a grows by adding 3 each time b grows by multiplying by 2 each time a) an = 3 + 4(n – 1) or an = 4n – 1 b) an = 3½ (2n) or an = 3 (2n-1) ©Evergreen Public Schools 2010

Sequences The sequence in the Rumors problem is a geometric sequence. It does not have a constant rate of change. Describe the data of change of the Rumors problem. Each is 4 times the last ©Evergreen Public Schools 2010

Geometric Sequences The sequences that are generated by repeated multiplication are called geometric sequences. 3, 6, 12, 24, 48, … from slide 12 is an example. The ratio is 2. ©Evergreen Public Schools 2010

Geometric Sequences 3, 6, 12, 24, 48, … from slide 12 is an example. Look for a pattern in ratio of consecutive terms Since the ratio is the same, we call it the common ratio. The ratio is 2. ©Evergreen Public Schools 2010

Equations of Geometric Sequences
We found the equations for the Rumors problem is b) What is the equation for 3, 6, 12, 24, … an=32n ©Evergreen Public Schools 2010

Find next term and write equation with sequence notation.
1, -10, 100, -1000, … II 5, 10, 20, 40, … III 4, 12, 36, 108, IV 64, 16, 4, 1, … I a5 =100010= an= 1(-10)n II a5 = 40  2 = 80 an= 52n III a5 = 108  3 = 324 an= 43n IV a5 = 1 (1/4)= 1/4 an= 64(1/4)n ©Evergreen Public Schools 2010

Find next term and write equation with sequence notation.
II -243, -81, -27, -9, … III IV 4, 6, 9, 13.5 I a5 = 1/16  1/2 = 1/32 an= (1/2)n II a5 = 9  1/3 = 3 an= 24(1/3)n III a5 = 1/8  1/2 = 1/16 an= 2(1/2)n IV a5 = 13.5  1.5 = an= 41.5n ©Evergreen Public Schools 2010

Write a formula to find the equations for geometric sequences
r = common ratio In explicit form an = n = term number In recursive form an = and an+1 = and an = a1  r(n – 1) an = r  an-1 and a1 =  an+1 = r  an and a1 =  ©Evergreen Public Schools 2010

Team Practice ©Evergreen Public Schools 2010

Debrief Complete the sequence organizer for geometric sequences. ©Evergreen Public Schools 2010