2. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Do Now: Aim: What are geometric sequences? Annie deposits $1000 in a bank at 8%. interest is compounded quarterly. How much does Annie earn after 1 year? Balance A, Principal P, rate r, # of compoundings n, time t Compound Interest 1000(1 + 0.02) 4 1 = 1082.43

3. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Money in the Bank 1,2,3 Years Annie deposits $1000 in a local bank at 8%. Interest is compounded annually. How Much does Annie earn after 1 year? Simple interest - paid only on the initial principal 1000 + 1000(0.08) = $1080 principal End of year balance interest earned interest rate How much does Annie earn after 2 years? $1000(1.08)(1.08) = 1166.40 How much does Annie earn after 3 years? $1000(1.08)(1.08) (1.08) = 1259.712 1166.40 1080 or 1000(1.08)

4. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Exponential Growth Principal(1 + interest rate) number of years = Ending balance for 3 years Post growth y, Pre-growth A rate r, time t y = a b x y = A(1 + r) t Exponential function recall: In general terms year 1 2 3 geometric sequence 1166.401259.712 amount $1080

5. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. positive integers1 2 3 4 · · · n Definition of Geometric Sequence A sequence is geometric if the ratios of consecutive terms are the same. Sequence a 1, a 2, a 3, a 4,..... a n,... is geometric if there is a number r, r  0, such that and so on. The number r is the common ratio of the geometric sequence. terms of sequence a1a1 a1ra1ra1r2a1r2 a1r3a1r3 a 1 r n - 1

6. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. 2, 4, 8, 16,...., 12, 36, 108, 324,....r = 3 r = 2 Definition of Geometric Sequence A sequence is geometric if the ratios of consecutive terms are the same. Sequence a 1, a 2, a 3, a 4,..... a n,... is geometric if there is a number r, r  0, such that and so on. The number r is the common ratio of the geometric sequence. r = ? n th ? r = -1/3 2 n,... 4(3) n,...

7. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. The n th term of an geometric sequence has the form where r is the common ratio between consecutive terms of the sequence. Thus ever geometric sequence can be written in the following form The n th Term of a Geometric Sequence a 1 a 2 a 3 a 4..... a n.... a 1 a 1 r a 1 r 2 a 1 r 3.... a n = a 1 r n – 1, a 1 r n - 1.... r0r0 2, 6, 18, 54 r = 3

8. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. n th Term of a Geometric Sequence Find the first five terms of the geometric sequence whose first term is a 1 = 3 and whose common ratio is r = 2. a n = a 1 r n – 1 a 1 = a 1 r 1 – 1 a 2 = a 1 r 2 – 1 a 3 = a 1 r 3 – 1 a 4 = a 1 r 4 – 1 a 5 = a 1 r 5 – 1 = 3(2) 0 = 3(2) 1 = 3(2) 2 = 3(2) 3 = 3(2) 4 = 3 = 6 = 12 = 24 = 48

9. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Finding a Term Find the 12 th term of the geometric sequence 5, 15, 45,... a n = a 1 r n – 1 a 12 = 5(3) 11 = 885,735 r = 15/5 = 3 Find the 15 th term of the geometric sequence whose first term is 20 and whose common ratio is 1.05. a n = a 1 r n – 1 a 15 = 20(1.05) 14  39.599 r = 1.05 The 4 th term of a geo. sequence is 125, and the 10 th is 125/64. Find the 14 th. 4 th term times r 6 = 10 th term or a 10 = a 4 r 6 Consider the 4 th term as if it were the 1 st term

10. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Model Problems Find the seventh term of the geometric progression 32, -16, 8,.... a n = a 1 r n – 1 a 7 = 32(-1/2) 6 = 1/2 r = -16/32 = -1/2

11. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Model Problems There are three number such that the second is two more than the first and the third is nine times the first. The numbers form a geometric sequence. Find the numbers. let x = 1 st number x + 2 = 2 nd number 9x = 3 rd number -½, 1½, -4½ 1, 3, 9

12. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Model Problem A company purchases $50,000 worth of copiers on Jan. 1, 1996. This asset depreciates at a rate of 45% per year. What is the value of this asset at the end of 2001 (or 1/1/02)? a n = a 1 r n – 1, (note: a 45% depreciation rate means the asset retains 55% of its value) a 1 = 50,000r =.55n = 2002... 1996 = 7 a 7 = 50000(.55) 7-1 a 7  1384.032031 Asset is worth approximately $1384.03

13. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Do Now: Aim: Is it possible to find the sum of a geometric sequence?

14. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Find the sum of the first eight terms of the geometric sequence 1, 3, 9, 27,... The Sum of a Finite Geometric Sequence r = ?3 a 1 a 2 a 3 a 4..... a n.... a 1 a 1 r a 1 r 2 a 1 r 3.... a 1 r n - 1.... 1 3 9 27....1(3) 7 1 + 3 + 9 + 27 +...+1(3) 7 = 3280 Is there an easier way to find the sum of a geometric sequence?

15. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Find the sum of the first eight terms of the geometric sequence 1, 3, 9, 27,... previous problem The Sum of a Finite Geometric Sequence The sum of the finite geometric sequence a 1, a 1 r 2, a 1 r 3, a 1 r 4,.... a 1 r n - 1.... with common ratio r  1 is given by r = 3 divergent series – the sum of the series is infinite when |r| > 1

16. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. The Sum of a Infinite Geometric Sequence Find the sum of 1 + 1/2 + 1/4 + 1/8 +... a 1 = 1 and r = ?0.5 S =? sum of the finite geometric sequence n = 4 1 + 1/2 + 1/4 + 1/8= 1 7/8 n = 5 1 + 1/2 + 1/4 + 1/8 + 1/16= 1 15/16 n = 6 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32= 1 31/32 n = 3 1 + 1/2 + 1/4= 1 3/4 as n   S approaches 2

17. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. The Sum of a Infinite Geometric Sequence Find the sum of 3 + 0.3 + 0.03 + 0.003 +... a 1 = 3 and r = ?0.1 3 + 3(0.1) + 3(0.1) 2 + 3(0.1) 3 +...

18. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Interest Problems Suppose at the beginning of each quarter you deposit $25 in a savings account that pays an APR of 2% compounded quarterly. Banks post interest at end of quarters. What would be the balance at year’s end? Date of Deposit 1 st year Additions Value at end of Quarter Jan 1 April 1 July 1 Oct 1 Account balance at end of year 25.50 25.38 25.25 25.13 $101.26 The sum represents a finite geometric series where a 1 = 25.13, r = 1.005 and n = 4

19. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Model Problems Find the sum of the series. 160 + 80 + 40 +..., n = 6 Find the sum of the infinite series. 160 + 80 + 40 +....

20. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Model Problems Find the sum of the geometric series.

21. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Model Problems Find a 1. S n = -55, r = -2/3, n = 5

22. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. A company purchases $50,000 worth of copiers on Jan. 1, 1996. This asset depreciates at a rate of 45% per year. What is the value of this asset at the end of 2001 (or 1/1/02)? a n = a 1 r n – 1, Model Problems

23. Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Find the nth term of the geometric sequence. a 2 = -18; a 5 = 2/3; n = 6 Model Problems