Essential Question: What is a sequence and how do I find its terms and sums?

How do I find the sum & terms of geometric sequences and series?

Geometric Sequences

Geometric Sequence– a sequence whose consecutive terms have a common ratio.

A sequence is geometric if the ratios of consecutive terms are the same.

Geometric Sequence

32 4

1 2 3

.....aa a

ra a a

The number r is the common ratio ratio.

2, 4, 8, 16, …, formula?, …

Ex. 1

12, 36, 108, 324, …, formula?, …

1, 4, 9, 16, …, formula? , …

Are these geometric?

1 1 1 1, , , ,..., ?,...

3 9 27 61formula

Yes 2n

Yes 4(3)n

No n2

No (-1)n /3

Finding the nth term of a Geometric Sequence

an = a1rn – 1

ra

a 2

1

Ex. 2bWrite the first five terms of the geometric sequence whose first term is a1 = 9 and r = (1/3).

9 3 11

3

1

9, , , ,

Ex. 3Find the 15Find the 15thth term of the geometric term of the geometric sequence whose first term is 20 and sequence whose first term is 20 and whose common ratio is 1.05whose common ratio is 1.05

an = a1rn – 1

a15 = (20)(1.05)15 – 1

a15 = 39.599

Ex. 4 Find a formula for the nth term.

What is the 9th term?

5, 15, 45, …

an = 5(3)n – 1

an = 5(3)n – 1

a9 = 5(3)8

a9 = 32805

an = a1rn – 1

sa r

rn

n

1 1

1

( )

sum of a finite geometric series

Ex. 6Find the sum of the first 12 terms of the Find the sum of the first 12 terms of the series 4(0.3series 4(0.3))nn

= 4(0.3)1 + 4(0.3)2 + 4(0.3)3 + … + 4(0.3)12

r

raaS

n

n

1

11

3.01

)3.0(2.12.1 12

12

S = 1.714

Ex. 7Find the sum of the first 5 terms of the Find the sum of the first 5 terms of the series 5/3 + 5 + 15 + …series 5/3 + 5 + 15 + …

r = 5/(5/3) = 3

r

raaS

n

n

1

11

31

)3(35

35 5

5

S = 605/3

What are your questions?

Homework:

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#16-22 even, 26, 27, 30, 32, 37, 38