Geometic Sequences Geometric sequences multiplied …

1 Algebra 2 Geometric Sequences and series Notes Mrs. Grieser Name: _____ Date: _____ Block: _____ Geometic Sequences Geometric Sequences contain a pattern where a fixed amount is multiplied from one term to the next (common ratio r) after the first term Geometric sequence examples: o 2, 4, 16, 32, .. o Domain: _____ o Range: _____ o Graph shown at right o common ratio r = _____ o The graph of a Geometric sequence is _____ o Find the common ratio (r) for the following Geometric Sequences : a) 5, 10, 20, 40, .. r = _____ b) -11, 22, -44, 88, .. r = _____ c) 4, 38, 916, 2732, 8164, .. r = _____ Identifying Geometric Sequences o Identify whether the following Sequences are arithmetic, Geometric , or neither.

2 If it is arithmetic, find d and if it is Geometric , find r. a) 4, 10, 18, 28, 40, .. _____ b) 625, 125, 25, 5, 1, .. _____ c) 81, 27, 9, 3, 1, .. _____ d) 1, 2, 6, 24, 120, .._____ e) -4, 8, -16, 32, -64, .._____ f) 8, 1, -6, -13, -20, .. _____ Algebra 2 Geometric Sequences and series Notes Mrs. Grieser Page 2 Finding Terms in a Geometric sequence o Find the 7th term in the sequence : 2, 6, 18, 54, .. r = _____ a7 = _____ o Is there a pattern? a1 = 2 a2 = a1 r a3 = a2 r = a1 r r = _____ a4 = a3 r = a2 r r = a1 r r r = _____ an = _____ o You a) Find the common ratio r: 6, -3, 23, 43.

3 B) Find the common ratio for the sequence given by the formula an=5(3)n-1 c) Find the 7th term of the sequence : 2, 6, 18, 54, .. d) Find a8 for the sequence , , , , .. e) Write a rule for the nth term of the sequence , then find a7 4, 20, 100, 500, .. f) One term of a Geometric series is a4=12. The common ratio r=2. Write a rule for the nth term. g) Two terms in a Geometric sequence are a3 = -48 and a6 = 3072. Find a rule for the nth term. To find the nth term in a Geometric sequence : an = a1 rn 1 where a1 is the first term of the sequence , r is the common ratio, n is the number of the term to find Algebra 2 Geometric Sequences and series Notes Mrs.

4 Grieser Page 3 Geometric series A Geometric series is the sum of the terms in a Geometric sequence : Sn = niira111 Sums of a Finite Geometric series o The sum of the first n terms of a Geometric series is given by: where a1 is the first term in the sequence , r is the common ratio, and n is the number of terms to sum. o Why? Expand Sn = _____ Multiply both sides by r: _____ Subtract: _____ Solve for Sn: _____ o Examples: a) Find the sum: 1611)3(4ii b) Find the sum of the first 8 terms of the sequence : -5, 15, -45, 135, .. c) Find the sum: 513kk o You a) Find the sum: 811)2(6kk b) Find the sum of the first 8 terms of the sequence : 6, 24, 96.

5 C) A soccer tournament has 64 participating teams. In the first round, 32 games are played. In each successive round, the number of games decreases by one half. Find a rule for the number of games played in the nth round, and the total number of games played. Sn = rran111 Algebra 2 Geometric Sequences and series Notes Mrs. Grieser Page 4 Sums of Infinite Geometric series Consider the series : ..321161814121 Is it a Geometric series ? _____ What is r? _____ Find the first 5 partial sums, S1, S2, S3, S4, and S5: o S1 = 21 o S2 = 4121 o S3 = 814121 _____ o S4 =161814121 = _____ o S5 = 321161814121_____ Graph these partial sums: What do you think will happen as we increase n?

6 _____ Examine the formula for the partial sum: Sn = rran111 o What happens as n gets very big (approaches infinity)? Consider values of r > 1 _____ r < -1 _____ -1 <r < 1 _____ An infinite Geometric series will converge if |r|<1; otherwise it will diverge Sum of an Infinite Geometric series Formula S = ra 11, when |r|< 1 Algebra 2 Geometric Sequences and series Notes Mrs. Grieser Page 5 Examples: Find the sum, if a) 11) (5ii b) ..6427169431 You the sum of the infinite series , if a) 1121kk b) 11453jj c) ..643163433 Recursive Formulas So far, we have worked with explicit formulas for arithmetic and Geometric Sequences o The explicit rule for the nth term of an arithmetic sequence : _____ o The explicit rule for the nth term of an Geometric sequence : _____ We can also define terms of a sequence recursively o Recursive formulas define one or more initial terms, and then each further term is defined as a function of preceding terms.

7 O Examples of recursion: Fibonacci sequence Initial terms: a1=0, a2=1 Recursive equation: an = an-1 + an-2 Expand: _____ Factorial function Initial terms: 0! = 1 Recursive equation: n! = n*(n-1)! (for n > 0) Expand: _____ Algebra 2 Geometric Sequences and series Notes Mrs. Grieser Page 6 Recursive formulas for arithmetic and Geometric Sequences o Recursive formula for arithmetic Sequences : _____ Add the common difference to the previous term o Recursive formula for Geometric Sequences : _____ Multiply the common ratio to the previous term o Examples: Write a recursive rule for the a) 3, 13, 23, 33, 43.

8 B) 16, 40, 100, 250, 625, .. c) Write the first 5 terms of the sequence : a1=3; an=an-1 - 7 o You a) Write the first 6 terms of the sequence : a0=1, an=an-1 + 4 b) Write the first 6 terms of the sequence : a1=1, an=3an-1 c) Write a recursive rule for the sequence : 2, 14, 98, 686, 4802, .. d) Write a recursive rule for the sequence : 19, 13, 7, 1, -5, .. e) Write a recursive rule for the sequence : 1, 1, 2, 3, 5, .. f) Write a recursive rule for the sequence : 1, 1, 2, 6, 24, .. Algebra 2 Geometric Sequences and series Notes Mrs. Grieser Page 7 Formula Summary: Sequences Explicit Recursive Arithmetic an = a1 + (n 1)d an = an-1 + d Geometric an = a1 rn 1 an = an-1 r series Sum of first n integers: nii1= 2)1( nn Sum of first n2 integers: nii12= 6)12)(1( nnn Sum of arithmetic series : Sn = niia1= 2)(1naan Sum of Geometric series : Sn = 11 iniira= rran 1)1(1, r 1 Sum of infinite Geometric series S = 11 iiira=ra 11, |r|<1