1. Warm Up
A petting zoo starts a population with one newborn male and one newborn female. Assuming that each adult pair will produce one male and one female offspring per month starting at two months, how many rabbits will there be after 6 months?

2. Geometric Sequences and Series
Keeper
Accelerated Pre-Calculus

3. Geometric Sequences
A sequence in which each term after the first is found by multiplying the preceding term by the same number is a geometric sequence. The number that is multiplied by each term to produce the next term is called the common ratio.

4. Example
For each of the following geometric sequences, identify the common ratio.
4, 16, 64, 256, 1024, …
, …
$750, $600, $480, $384, …

5. nth Term of a Geometric Sequence
The nth term of a geometric sequence is given by the formula:
an = a1r n  1
Example: Find the 11th term of the geometric sequence
1, 3, 9, 27, …

6. Example
Write a rule for the nth term of the geometric sequence that has the two given terms. 𝒂 𝟒 =𝟒𝟎 𝒂 𝟖 =𝟔𝟒𝟎

7. Write an explicit and recursive formula for the sequence
36, 12, 4, …

8. Write an explicit and recursive formula for the sequence
−2,10,−50,…

9. Write an explicit and recursive formula for the sequence
4, 8, 16, …

10. Write an explicit and recursive formula for the sequence
15, 5, 5 3 , …

11. Write an explicit and recursive formula for the sequence
1 32 , 1 16 , 1 8 ,…

12. Find the desired term
𝑎 9 for 60, 30, 15, …

13. Find the desired term
𝑎 5 for 3, 1, 1 3 , …

14. Find the desired term
𝑎 7 if a 3 =24 and r= 1 2

15. Find the desired term
𝑎 6 if a 1 =16,807 and r= 3 7

16. Find the desired term
𝑎 6 if a 1 =9 and r=2

17. If 𝑟=4 and 𝑎 8 =100, what is the first term of the geometric sequence?
Find the desired term
If 𝑟=4 and 𝑎 8 =100, what is the first term of the geometric sequence?

18. Sum of the First n Terms
The formula for the sum of the first n terms of a geometric sequence is given by: Example: Find the sum:

19. Find the sum of the geometric series
First six terms of …

20. Find the sum of the geometric series
First nine terms of (−1)+2+…

21. Find the sum of the geometric series
First eight terms of …

22. Find the sum of the geometric series
First 𝑛 terms of 𝑎 1 =4, 𝑎 𝑛 =2000, 𝑟=−3

23. Find the sum of the geometric series
First 𝑛 terms of 𝑎 1 =5, 𝑎 𝑛 =1,747,625, 𝑟=4

24. Find the sum of the geometric series
First 𝑛 terms of 𝑎 1 =3, 𝑎 𝑛 =46,875, 𝑟=−5

25. Find the sum of the geometric series
First 𝑛 terms of 𝑎 1 =8, 𝑎 𝑛 =−256, 𝑟=2

26. Find the sum of the geometric series
First 𝑛 terms of 𝑎 1 =−36, 𝑎 𝑛 =972, 𝑟=7

27. Find the Sum
𝑛= 𝑛−1

28. Find the Sum
𝑛=1 5 −3 𝑛−1

29. Find the Sum
𝑛= 𝑛−1

30. Find the Sum
𝑛= − 𝑛−1

31. Find the common ratio
𝑎 3 =12, 𝑎 6 =187.5

32. Find the common ratio
𝑎 2 =−6, 𝑎 7 =−192

33. Find the common ratio
𝑎 4 =−28, 𝑎 6 =−1372

34. Find the common ratio
𝑎 5 =6, 𝑎 8 =−0.048

35. Find the missing quantity for the geometric sequence
Find 𝑎 1 if 𝑆 12 =1365 and 𝑟=2

36. Find the missing quantity for the geometric sequence
Find 𝑎 6 if 𝑆 6 = , 𝑎 1 =100 and 𝑟= 1 2

37. Find the missing quantity for the geometric sequence
Find 𝑟 if 𝑎 1 =0.12, 𝑆 𝑛 =590.52, and 𝑎 𝑛 =787.32

38. Find the missing quantity for the geometric sequence
Find 𝑛 for … if 𝑆 𝑛 =61.5

39. Write in sigma notation
…+3072

40. Write in sigma notation
…+1152

41. Write in sigma notation…

42. Write in sigma notation
1 8 − −…+8

43. Write in sigma notation
0.2−1+5−…−625

44. Example
Marc is making monthly payments on a loan. Suppose instead of the same monthly payment, the bank requires a low initial payment that grows at an exponential rate each month. The total cost of the loan is represented by 𝑛=1 𝑘 𝑛−1
What is Marc’s initial payment and at what rate is this payment increasing?
If the sum of Marc’s payments at the end of the loan is $7052, how many payments did Marc make?