The document discusses geometric sequences and provides examples of finding terms in geometric sequences using the formula an = a1r(n-1) where a1 is the first term, r is the common ratio, and n is the term number. It gives the example of finding the three geometric means between the terms 12 and 3072 in the sequence 12, 48, 192, 768, 3072, which has a common ratio of 4.
1. What comes next? 3, 6, 12, __, __, __ A geometric sequence is a pattern of numbers in which each number is found by multiplying a constant by the number before it. 24 48 96 x2 x2 x2 x2 x2 x2 Common ratio r = 2 a1 = 3 a2 = 6 a3 = 12
2. What’s the common ratio? 8, 4, 2, 1, … r = 0.5 2, -6, 18, -54, … r = -3 Alternating sequence
3. 3, 6, 12, 24, 48, 96, … x2 x2 x2 a1 = 3 a2 = 6 3 x 21 a3 = 12 3 x 22 a4 = 24 3 x 23 an = a1 x r (n – 1)
4. Write an equation for the nth term of the geometric sequence 7, 14, 28, …. an = a1 r (n – 1) an = 7 2(n-1) Find the 80th term of the sequence an = 7 2 (n – 1) a80 = 7 279 Example 1-3a
5. Complete the statement for the geometric sequence: 1536 is the ____ term of 3, 6, 12, … an = a1 r (n – 1) 1536 = 3 2 (n – 1) 512 = 2(n – 1) 29 = 2(n – 1) 9 = n – 1 n = 10 ____ _ 3 3
6. The terms between any two nonconsecutive terms of a geometric sequence are geometric means. Find the three geometric means between 12and 3072. 12, ___, ___, ___, 3072, ..., Example 1-4a
7. The terms between any two nonconsecutive terms of a geometric sequence are geometric means. Find the three geometric means between 12and 3072. 12, ___, ___, ___, 3072, ..., xr xr xr xr Example 1-4a
8. The terms between any two nonconsecutive terms of a geometric sequence are geometric means. Find the three geometric means between 12and 3072. 12, ___, ___, ___, 3072, ..., xr xr xr xr 12 x r4 = 3072 r4 = 256 r = 4 Example 1-4a
9. The terms between any two nonconsecutive terms of a geometric sequence are geometric means. Find the three geometric means between 12and 3072. 48 192 768 12, ___, ___, ___, 3072, ..., xr xr xr xr 12 x r4 = 3072 r4 = 256 r = 4 Example 1-4a
