7.2 GEOMETRIC Sequences

Homefun  pg 430-432 # 1,2,3, 4,5abd, 6bdef, 7def, 9cd, 11, 12

REVIEW Sequence: an ordered list of numbers Term: a number in a sequence (the first term is referred to as t1, the second term as t2, etc…) example 3, 7, 11, 15, … t1 = 3 t2 = 7 t3 = 11 t4 = 15

REVIEW - Recursive SEQUENCE a sequence for which one or more terms are given each successive term is determined by performing a calculation using the previous term(s) example t1 = 2 describes 2, 6, 18, 54, … tn =3 tn-1 t2 =3t1 =3(2) = 6 n>1 , n  N t3 =3t2 =3(6) = 18

REVIEW - General term a formula that expresses each term of a sequence as a function of its position labelled tn example tn = 2n describes 2, 4, 6, 8, 10

REVIEW - arithmetic SEQUENCE a sequence that has a common difference between any pair of consecutive terms The general arithmetic sequence is a, a + d, a + 2d, a + 3d, …, where a is the first term and d is the common difference. example 3, 7, 11, 15, … has a common difference of 4 7 – 3 = 4 11 – 7 = 4 15 – 11 = 4

REVIEW - Arithmetic sequence General term Recursive formula tn = a + (n – 1)d where a is the first term d is the common difference n  N t1 = a tn = tn-1 + d n > 1 , n  N

REVIEW - Arithmetic sequence DISCRETE Linear function f (n) = dn + b where b = t0 = a - d

GEOMETRIC SEQUENCE a sequence that has a common ratio between any pair of consecutive terms the general geometric sequence is a, ar, ar2, ar3, …, where a is the first term and r is the common ratio example 2, 6, 18, 54, … has a common ratio of 3 62= 3 18  6 = 3 54  18 = 3

t1 = a tn = rtn-1 n > 1 , n  N tn = ar n-1 GEOMETRIC SEQUENCES General term Recursive formula tn = ar n-1 where a is the first term r is the common ratio n  N t1 = a tn = rtn-1 n > 1 , n  N

GEOMETRIC SEQUENCES DISCRETE EXPONENTIAL FUNCTION f(n) = ar n-1

Example 1 State the common ratio and write the next three terms : a) 4, 8, 16, … b) -7.8, 3.8, –1.9, … c)

Example 2 tn = ar n-1 t1 = a, tn = rtn-1 n > 1 For each sequence, determine the general term, the recursive formula and the indicated term. 5, 10, 20, … , t9 a, – 2ab, 4ab2, … , t17

Example 3 tn = ar n-1 t1 = a, tn = rtn-1 n > 1 Find the number of terms in each of the following geometric sequences. 3, 6, 12, … , 768 567, 189, 63, … , 7/27

Example 4 tn = ar n-1 t1 = a, tn = rtn-1 n > 1 The 3rd term of a geometric sequence is 2 and the 10th term is . Determine the general term.

Homefun  pg 430-432 # 1,2,3, 4,5abd, 6bdef, 7def, 9cd, 11, 12