1. GEOMETRIC SEQUENCES

2. 10, 20, 40, 80, …….. ÷2 This is called a common ratio. (Term 2 is divided by term 1. Term 3 is divided by term 2 and so on.) This makes it a geometric sequence. This could be called a “exponential sequence”. Consider this sequence. Notice that to go from one term to the next multiplication is required. Defined by the formula: a n = a 1 (r) n-1

3. WARNING: Don’t confuse geometric with sequences that increase through squaring or cubing. You won’t have a problem as long as you check several quotients when looking for the common ratio.

4. Find the 15 th term of the geometric sequence whose first term is 20 and whose common ratio is 1.05. You could write out the first 15 terms or use the formula. a n = a 1 (r) n-1 a 15 = 20 (1.05) 15-1

5. Find the 12 th term of the geometric sequence 5, 15, 45,…… After we find the common ratio we can just use the formula. a n = a 1 (r) n-1 a 12 = 5 (3) 12-1

6. The fourth term of a geometric sequence is 125 and the 10 th term is 125/64. Find the 14 th term First we have to play with this formula. a n = a 1 (r) n-1 Isn’t it true that for example if we had. a 10 = a 1 (r) 10-1 a 10 = a 1 (r) 9

7. And isn’t it true that we COULD rewrite this as…. a 10 = a 1 (r) 9 a 10 = a 1 r·r·r·r 6

8. X X X X X X And that leaves us with a relationship that we can use on this question. a 10 = a 4 r 6

9. And that leaves us with a relationship that we can use on this question. a 10 = a 4 r 6 Notice that the subscript and the exponent on the right add up to the subscript on the left. The fourth term of a geometric sequence is 125 and the 10 th term is 125/64. Find the 14 th term

10. Substituting and solving for r we find…. 125/64 = 125 r 6 1/64 = r 6 ½ = r The fourth term of a geometric sequence is 125 and the 10 th term is 125/64. Find the 14 th term

11. Substituting and solving for r we find…. 125/64 = 125 r 6 1/64 = r 6 ½ = r The fourth term of a geometric sequence is 125 and the 10 th term is 125/64. Find the 14 th term

12. NOW we can answer the question a 14 = a 10 r 4 a 14 = 125/64( ½) 4 a 14 = 125/1024 The fourth term of a geometric sequence is 125 and the 10 th term is 125/64. Find the 14 th term

15. Finding the SUM of a FINITE GEOMETRIC SEQUENCE