( DAY 2) V OCABULARY Two types of sequences were studied: Arithmetic - PowerPoint PPT Presentation

D AY 92 – G EOMETRIC SEQUENCES ( DAY 2)

V OCABULARY Two types of sequences were studied: Arithmetic Sequence: A sequence is called arithmetic if there is a real number d such that each term in the sequence is the sum of the previous term and d. Geometric Sequence: A sequence is called geometric if there is a real number r such that each term in the sequence is a product of the previous term and r.

List the first five terms of each sequence, and identify them as arithmetic or geometric.   A(n+ 1 )= A(n)+ 4 for n 1 and A( 1 )= 2 1. 1  A(n+ 1 )= *A(n)for n 1 and A( 1 )= 8 2. 4

List the first five terms of each sequence, and identify them as arithmetic or geometric.   1. A(n+ 1 )= A(n)+ 4 for n 1 and A( 1 )= 2 - 2,2,6,10,14 Arithmetic 1  1 1 1 8 A(n+ )= *A(n)for n and A( )= 2. 4 Geometric 1 1 1 8 , 2 , , , 2 8 32

   3. A(n+ 1 )= A(n) 19 for n 1 and A( 1 )= 6 2  4. 1 1 1 6 A(n+ )= A(n)for n and A( )= 3

   3. A(n+ 1 )= A(n) 19 for n 1 and A( 1 )= 6 – 6, – 25, – 44, – 63, – 82 Arithmetic 2  4. 1 1 1 6 A(n+ )= A(n)for n and A( )= 3 8 16 32 6 , 4 , , , Geometric 3 9 27

N TH TERM OF AN G EOMETRIC S EQUENCE Every geometric sequence can be defined using the explicit formula:   1 3 n 1 a a r n n 1 Where: a n n th term a 1 1 st term n term number r common ratio

W ORD P ROBLEM 1 A rabbit population grew in the following pattern: 2, 4, 8, 16, . . . If all the rabbits live and the pattern continues, how many rabbits will be in the 8 th generation?

S OLUTION   n 1 a a r n 1  a 2 1  r 2 - doubling  n 8   8 1 a 2 * ( 2 ) 8  256 rabbits

P ROBLEM Consider the equation y = 150(2 x ) a. Make a table x and y – values for whole-number x-values from 0 to 5. x 0 1 2 3 4 5 y b. What do the numbers 150 and 2 in the equation tell you about the relationship?

A NSWER K EY Consider the equation y = 150(2 x ) a. Make a table x and y – values for whole-number x-values from 0 to 5. x 0 1 2 3 4 5 y 150 300 600 1200 2400 4800 b. What do the numbers 150 and 2 in the equation tell you about the relationship? 150 is the initial value 2 is the growth rate

W ORD P ROBLEM 2 Fido did not have fleas when his owners took him to the kennel. The number of fleas on Fido after he returned from the kennel grew according to the equation f = 8(3 n ). Where f is the number of fleas returned from the kennel. (Fido left the kennel at week 0.) a. How many fleas did Fido pick up at the kennel? b. What is the growth factor for the number of fleas? c. How many fleas will Fido have after 10 weeks if he is not treated?

A NSWER K EY Fido did not have fleas when his owners took him to the kennel. The number of fleas on Fido after he returned from the kennel grew according to the equation f = 8(3 n ). Where f is the number of fleas returned from the kennel. (Fido left the kennel at week 0.) a. How many fleas did Fido pick up at the kennel? 8 b. What is the growth factor for the number of fleas? 3 c. How many fleas will Fido have after 10 weeks if he is not 8(3 10 ) = 472,392 treated?