12.3 Analyze Geometric Sequences and Series

12.3 Analyze Geometric Sequences and SeriesGoal • Study geometric sequences and series.VOCABULARYGeometric sequence________________________________________________________________________________________________________________________________________________Common ratio________________________________________________________________________________________________________________________________________________Example 1Identify geometric sequences.Tell whether the sequence 1,4,16, 64, 256,… is geometric.To decide whether a sequence is geometric, find the ratios of consecutive terms.a 4 a 12 14a32= _____ = _____a4aa43= ______ = ____a 5a= ______ = ______Each ratio is ____, so the sequence _____ geometric.Checkpoint Tell whether the sequence is geometric.512, 128, 64, 8,…RULE FOR A GEOMETRIC SEQUENCEThe nth term of a geometric sequence with first term a 1 and common ratio r is given by:a n = a 1 r n1

Example 2Write a rule for the nth termWrite a rule for the nth term of the sequence 972, 324,108, 36, …. Then find a 10SolutionThe sequence is geometric with first term a 1 = _____r = _______ = _____. So, a rule for the nth term is:a n = a 1 r n1Write general rule.and common ratio n 1= _____ ____ Substitute for a 1 and r.The 10th term is a 10 = ______________ = __________Checkpoint Write a rule for the nth term of the geometric sequence. Find a 9 .14, 28, 56, 112,…Example 3Write a rule given two termsTwo terms of a geometric sequence are a 2 = 10 and a 7 = 320. Find a rule for the nthterm.1. Write a system of equations using a n = a 1 r n1 and substituting 2 for n (Equation 1)and then 7 for n (Equation 2).a 2 = a 1 r 21 Equation 1a 7 = a 1 r 71 Equation 22. Solve the system.3. Find a rule a n . a n = a 1 r n1 Write general rule.a n =Substitute.

Checkpoint Write a rule for the nth term of the geometric sequence. Find a 9 .a 3 = 224, a 6 = 2812.5 Use Recursive Rules with Sequences and FunctionsVOCABULARYExplicit rule______________________________________________________________________________________________________________________________________________Recursive rule______________________________________________________________________________________________________________________________________________Example 1Evaluate recursive rulesWrite the first six terms of the sequence.a 0 = 2, a n = a n 1 3Solutiona 0 = 2a 1 = a 0 3 = ____ = ____a 2 = a 1 3 = ____ = ____a 3 = a 2 3 = ____ = ____a 4 = a 3 3 = ____ = ____a 5 = a 4 3 = ______ = ____Checkpoint Write the first five terms of the sequence.a 0 = 4, a n = 1.5a n1

RECURSIVE EQUATIONS FOR ARITHMETIC AND GEOMETRICSEQUENCESArithmetic Sequence a n = a n1 + d where d is the common differenceGeometric Sequence a n = r a n1 where r is the common ratioExample 2Write recursive rulesWrite a recursive rule for the sequence.a. 1, 7, 13, 19, 25, . . . b. 4, 12, 36, 108, 324, . . .Solutiona. The sequence is ___________ with first terma 1 =___ and common difference d = _____ = ___.a n = a n1 + d General recursive equation for a n= Substitute for d.So, a recursive rule for the sequence is a 1 = ___,a n = a n1 +6b. The sequence is ___________ with first term a 1 = ___and common ratio r = _____=___a n = r a n1 General recursive equation for a n .= Substitute for r.So, a recursive rule for the sequence is a 1 = ___,a n = 3a n1