Arithmetic & Geometric Sequences

Vocabulary: sequence: a function whose domain is the set of consecutive integers greater than or equal to k. (usually k = 1). Arithmetic Sequences – a sequence in which the difference between consecutive terms is constant. (add or subtract the same value)

DEFINITION OF A SEQUENCE An infinite sequence is a function whose domain is the set of positive integers. The function values, written as a1, a2, a3, a4, … , an, …, are called the terms of the sequence. The nth term, an, is called the general term of the sequence.

Formulas for Arithmetic Sequences Explicit Formulas – formula which shows the nth term of a sequence in terms of n. nth term constant difference first term

Formulas for Arithmetic Sequences Recursive Formulas – formula which the first term or first few terms are given, and then the nth tem is expressed using the preceding term(s). first term previous term constant difference nth term

Example 1: Write a recursive and explicit formula for the following arithmetic sequences: 1, 5, 9, 13, 17, 21.... 5, -1, -7, -13, .....

Vocabulary Geometric Sequences – a sequence in which the ratio of consecutive terms is constant. (multiply or divide by the same value)

Formulas for Geometric Sequences Explicit Formulas – formula which shows the nth term of a sequence in terms of n. nth term constant ratio first term

Formulas for Geometric Sequences Recursive Formulas – formula which the first term or first few terms are given, and then the nth tem is expressed using the preceding term(s). first term previous term nth term constant ratio

Example 2: Write a recursive and explicit formula for the following geometric sequences: 3, 6, 12, 24, 48, .... 4, -12, 36, -108, .....

Example 3: Find the 49th term in the arithmetic sequence 8, 15, 22, 29, … Difference = Explicit: an = Recursive:

Example 4: Give the 7th term in the geometric sequence 16, 24, 36, … Ratio = Explicit: gn = Recursive:

Closure: A particular car depreciates 25% in value each year. Suppose the original cost is $14,800. a) Find the value of the car in its second year (ie. after 1 year).  b) Write an explicit formula for the value of the car in its nth year.   c) In how many years will the car be worth about $1000?