Geometric Sequences & Exponential Functions

Why "Geometric" Sequence? a line is 1-dimensional and has a length of r in 2 dimensions a square has an area of r2 in 3 dimensions a cube has volume r3 etc (yes we can have 4 and more dimensions in mathematics).

Vocabulary for Understanding sequence term of a sequence common ratio infinite sequence finite sequence recursive formula explicit formula

Vocabulary … A sequence is an ordered set of numbers. Each number in the sequence is a term of the sequence. A sequence may be an infinite sequence that continues without end, such as the natural numbers, or a finite sequence that has a limited number of terms, such as {1, 2, 3, 4}.

You can think of a sequence as a function with sequential natural numbers as the domain and the terms of the sequence as the range. Values in the domain are called term numbers and are represented by n. Instead of function notation, such as a(n), sequence values are written by using subscripts. The first term is a1, the second term is a2, and the nth term is an. Because a sequence is a function, each number n has only one term value associated with it, an.

Putting it together to create a table Term Number Value n 1 2 3 4 5 … an 16 -8 -2 Domain Range an is read “a sub n.” …means sequence goes on forever −8 16 4 −8 −2 4 1 −2 Common ratio is - 1 2 as each ratio above simplifies to this value Because this sequence behaves according to the rule of multiplying a constant number to one term to get the next, this is called a geometric sequence. The fixed number that binds each sequence together is called the common ratio. ‘r’ represents the common ratio in the geometric sequence formula. NOTE: a fractional ratio causes the sequence to decay while a whole number ratio causes the sequence to grow.

Example: Sequence: 2, 4, 8, 16, 32, 64, 128, 256, ... This sequence has a factor of 2 (common ratio) between each number. In a table this geometric sequence would be: Each term (except the first term) is found by multiplying the previous term by 2. n 1 2 3 4 5 6 7 8 … an 16 32 64 128 256 Geometric Sequences are sometimes called Geometric Progressions (G.P.’s)

Geometric Sequence Formula n – term number you are looking for a1 – first number in the sequence r – common ratio From previous example To find the 12th term: a12 = 2(2)12-1 = 4096 for the given sequence.

Recursive vs. Explicit A recursive formula is a rule in which one or more previous terms are used to generate the next term. In some sequences, you can find the value of a term when you do not know its preceding term. An explicit formula defines the nth term (future term) of a sequence as a function of n.

Relationship Between Recursive and Explicit Formula’s Recursive Formula Explicit Formula an = an-1(r) an = a1(r)n-1 (next term) (future term)

Relationship Between Sequences and Functions Linear Functions : common difference (Arithmetic Sequence) Quadratic Functions : constant second difference (1ST/2ND Difference) Exponential Functions: constant ratios (Geometric Sequence)

Write a possible explicit rule for the nth term of the sequence.

Physics Example A ball is dropped and bounces to a height of 4 feet. The ball rebounds to 70% of its previous height after each bounce. Graph the sequence and describe its pattern. How high does the ball bounce on its 10th bounce? Create a table of input/output values for this example.

Physics Example Cont.…

Physics Example Cont.… The graph appears to be exponential. Use the pattern to create an input/output table. n 1 2 3 4 5 2.8 1.96 1.372 .9604 an a2 = 0.7(4) = 2.8 a3 = 0.7(2.8) = 1.96 a4 = 0.7(1.96) = 1.372 a5 = 0.7(1.372) = .9604 an = 4(0.7)n – 1, where n is the bounce number