Geometric Sequences A geometric sequence is a sequence

Geometric Sequences �A geometric sequence is a sequence where each # in the seq. is multiplied by a constant (which is called the common ratio (r)) �To find r, divide any term in the sequence by the previous term

Geometric Sequences �General Formula:

Geometric Sequences �Find the 11 th term of the geo. Sequence listed below � 64, -32, 16, -8, …a 11

Geometric Sequences �Find the 6 th term of the geo. sequence listed below � 3, -15, 75, . . a 6

Geometric Sequences �Write an equation for the nth term � 3, 12, 48, 192. . . an

Geometric Sequences �Find the 10 th term of the sequence if a 4=108 r=3

Geometric Sequences �Find the 7 th term of the sequence if a 3=96 r=2

Geometric Means �Geometric means are the missing terms between two non-successive terms in a geo. Sequence �Find 3 geometric means between 2. 25 and 576

Geometric Means Find 5 geometric means between ½ and 1/1458

Geometric Series �A series that is associated with a geometric sequence

Geometric Series �Find the sum of the first 6 terms of the geometric series 3 + 6 + 12 + 24 +…

Geometric Series �Find the first term of the series if the S 8=39, 360 and r=3

Geometric Series �Find the sum of the first 8 terms of 1+x+x 2+x 3+…

Sigma Notation �More concise (less time consuming) notation for writing out a series

Sigma Notation

Sigma Notation

Sigma Notation

Write in sigma notation � 1 + 3 + 5 + 7

Write in sigma notation � 2 + 4 + 6 + 8 + 10

Write in sigma notation � 3 + 6 + 12 + 24 + 48

Write in sigma notation �-3 + 9 + -27 + 81 + -243

Write in sigma notation �-2 + 4 + -8 +. . . +256

Infinite Geometric Series �In an infinite series, Sn approaches some limit as n becomes very large. That limit is defined to be the sum of the series. If an infinite series has a sum, it is said to converge. �A series converges (or has a sum) if and only if lrl < 1

Does the geom. series have a sum?

Does the geom. series have a sum?

Does the sum of each term approach some limit?

To find the sum of an infinite series �Make sure a limit exists first

An infinite series in sigma notation—find the sum

An infinite series in sigma notation—find the sum

Writing a repeating decimal as a fraction

Writing a repeating decimal as a fraction

A different kind of sequence. . . �

Expand �

Pascal’s Triangle 1 11 121 1331 14641 1 5 10 10 5 1 1 6 15 20 15 6 1 First row is used for anything to the zero power. Used for the coefficients of each term of the expanded binomial

Expand using Pascal’s Triangle

Expand using Pascal’s Triangle