11 3 Geometric Sequences What is a Geometric

11. 3 – Geometric Sequences

What is a Geometric Sequence? u In a geometric sequence, the ratio between consecutive terms is constant. This ratio is called the common ratio. u Unlike in an arithmetic sequence, the difference between consecutive terms varies. u We look for multiplication to identify geometric sequences.

Ex: Determine if the sequence is geometric. If so, identify the common ratio u 1, -6, 36, -216 yes. Common ratio=-6 u 2, 4, 6, 8 no. No common ratio

Important Formulas for Geometric Sequence: u Recursive Formula an = (an – 1 ) r Where: an is the nth term in the sequence a 1 is the first term n is the number of the term r is the common ratio u Explicit Formula an = a 1 * r n-1 u Geometric Mean Find the product of the two values and then take the square root of the answer.

Let’s start with the geometric mean u Find the geometric mean between 3 and 48 Let’s try one: Find the geometric mean between 28 and 5103

Ex: Write the explicit formula for each sequence First term: a 1 = 7 Common ratio = 1/3 Explicit: an = a 1 * r n-1 Now find the first five terms: a 1 a 2 a 3 a 4 a 5 = = = 7(1/3) 7(1/3) (1 -1) (2 -1) (3 -1) (4 -1) (5 -1) = = = 7 7/3 7/9 7/27 7/81

Explicit Arithmetic Sequence Problem Find the 19 th term in the sequence of 11, 33, 99, 297. . . an = a 1 * r n-1 Common ratio = 3 Start with the explicit sequence formula Find the common ratio between the values. a 19 = 11 (3) (19 -1) a 19 = 11(3)18 =4, 261, 626, 379 Plug in known values Simplify

Let’s try one Find the 10 th term in the sequence of 1, -6, 36, -216. . . an = a 1 * r n-1 Start with the explicit sequence formula Common ratio = -6 a 10 = 1 (-6) (10 -1) a 10 = 1(-6)9 = -10, 077, 696 Find the common ratio between the values. Plug in known values Simplify