MATH 110

1. MATH 110 EXAM 4 Review

2. Arithmetic sequence Geometric Sequence Sum of an Arithmetic Series Sum of a Finite Geometric Series Sum of Infinite Geometric Series

3. Jeopardy

4. Leftovers !? 100 • A culture of bacteria originally numbers 500 spores. After 2 hours there are 1500 bacteria. Assuming the number of spores can be modeled by the exponential function determine how many spores will be present in 6 hours. • Answer: 13,500 spores

5. Leftovers !? 200 • The cost of tuition at four-year public universities has been increasing roughly exponentially for the past several years. In 1997, average tuition was $3,111 while in 2004 it was $5,132. Assuming that tuition will increase according to the exponential model , at what rate is tuition increasing each year. • Answer: 7.41%

6. Leftovers !? 300 • When rabbits were first brought to Australia, they multiplied very rapidly because there were no predators. In 1865, there were 60,000 rabbits. By 1867, there 2,400,000 rabbits. Assuming exponential growth according to the model , when was the first pair of rabbits introduced into the country? • Answer: Around 1859

7. Leftovers !? 400 • The consumer price index compares the cost of goods and services over various years. The base year for comparison is 1967. The same goods and services that cost $100 in 1967 cost $184.50 in 1977. Assuming that costs increase exponentially according to , when did the same goods and services cost double that of 1967? • Answer: 1978

8. Leftovers !? 500 • The proportion of carbon-14, an isotope of carbon, in living plant matter is constant. Once a plant dies, the carbon-14 in it begins to decay with a half-life of 5570 years. An archaeologist measures the remains of carbon-14 in a prehistoric hut and finds it to be one-tenth the amount of carbon-14 in the living wood. How old is the hut? • Answer: 18,503 years old

9. Number Patterns 100 • Consider the following sequence: 7, 11, 15, 19, . . . Find the 150 term. • Answer: 603

10. Number Patterns 200 • How many terms are there in the following sequence: • Answer: 25 terms

11. Number Patterns 300 • List the first four terms for the sequence whose formula is: • Answer: 5, (1/5), 5, (1/5)

12. Number Patterns 400 • Consider the following sequence: 375, -75, 15, -3, . . . • What is the sign of the 493 term? • Determine the most evident formula for the nth term of the sequence. • Answer: positive;

13. Number Patterns 500 • Write a formula for the following sequence: • 1, 3, 6, 10, 15, . . . • Answer:

14. “Sum”thing 100 • Evaluate the expression: • Answer: 66

15. “Sum”thing 200 • Re-index the following summation so that it starts at k = 1:

16. “Sum”thing 300 • It can be shown that the Euler number, e, can be approximated taking the square root of the following series: • Write this series using sigma notation. • Answer:

17. “Sum”thing 400 • Find the first term given that and • Answer: 25

18. “Sum”thing 500 • Find the value of the sum: • Answer: 337,274

19. Two Steps Back 100 • Find the common difference of an arithmetic sequence whose 16th term is -73 and whose 21st term is -103. • Answer: -6

20. Two Steps Back 200 • Find the twentieth term of the arithmetic sequence whose third term is 6 and whose sixth term is 18. • Answer: 74

21. Two Steps Back 300 • Which of the following sequences is arithmetic? (I) (II) (III)

22. Two Steps Back 400 • Find the sum of the first 30 terms in an arithmetic sequence where the 6th term is 7 and the 12th term is 12. • Answer: 447.5

23. Two Steps Back 500 • Find the sum of all the numbers in the sequence: • Answer: 18980

24. Giant Leaps Forward 100 • Find the common ratio of a geometric sequence where the 4th term is 100 and the 7th term is 4/5. • Answer: 1/5

25. Giant Leaps Forward 200 • Find the first term of a geometric sequence whose fourth term is -8 and whose tenth term is -512/729. • Answer: -27

26. Giant Leaps Forward 300 • Which of the following sequences is geometric? (I) (II) (III)

27. Giant Leaps Forward 400 • Find the indicated sum: • Answer: 56/3

28. Giant Leaps Forward 500 • A company plans to contribute to each of its employees’ retirement fund by depositing $100 at the end of each month in a retirement account. The account pays 6% interest compounded monthly. A look at the account balance shows that the amount is a series: • How much money will there be after 18 years • Answer: 38,929

29. Potent Potables 100 • Find the sum: • Answer: 1661

30. Potent Potables 200 • Find the sum of the infinite geometric series if possible: • Answer: Not possible

31. Potent Potables 300 • A repeating decimal can always be expressed as a fraction. Consider the decimal: 0.23232323… Use the fact that: 0.23232323… = 0.23 + 0.0023 + 0.000023 + …. To write 0.232323… as a fraction. • Answer: 23/99

32. Potent Potables 400 • This problem illustrates how banks create credit and can lend out more money than has been deposited. Suppose that $100 is deposited in a bank. Experience shows that on average on 8% of the money deposited is withdrawn by the owner, which means that bank are free to lend 92% of their deposits. Thus, $92 of the original $100 is loaned out to other customers. This $92 will become someone else’s income, and eventually will be redeposited in the bank. So $92(0.92) =$84.64 is loaned out again and then redeposited, and so on. Find the total amount of money deposited in the bank. • Answer: $1250

33. Potent Potables 500 • Solve for x: • Answer: 1/2