Unit 5 Study Guide Name Objectives: Write the explicit form of arithmetic and geometric sequences and find the n th term. (Memorize those formulas.) Use formulas (given) to find the sum of the first n terms of a series. Write and evaluate sums in summation notation. Determine if a geometric sequence converges or diverges. Find the sum of an infinite series when possible. Arithmetic sequences have a common and geometric sequences have a common. Explicit form (n th term) Sum of the first n terms Sum of infinite series Arithmetic Sequences Geometric Sequences The term converging only applies to sequences. How do you determine if a sequences is converging or not? Examples 1. Given the sequence: 12, 15, 18, 21, a) Write the explicit formula. b) Find the 40 th term. c) Find the sum of the first 20 terms. d) The number 60 is what term in the sequence?
2. Given the sequence: 2, 8, 32, 128, a) Write the explicit formula. b) Find the 15 th term. c) Find the sum of the first 15 terms d) The number 2048 is what term in the sequence? 3. Determine if the series converges. If so, find the sum of the infinite series. a) 6561 + 2187 + 729 + b) -2 + 4 + (-8) + 16 + c) 70 + 65 + 60 + d) 80 + 40 + 20 + 10 + Expand and evaluate each sum. 4. 5. Write each series in summation notation. 6. -2 + 6 18 + 54 162 7. 8 + 12 + 16 + 20 + 24 + 28 + 32 8. Once a week Mrs. Baker makes sugar cookies. The first week she makes the recipe, she uses the full 2 cups of sugar called for by the recipe. Each week after that, she reduces the amount of sugar by one third. A. How much sugar does she use for the cookies on the fifth week? B. How much sugar does she use for cookies over half a year? (There are 52 weeks in a year.) C. If Mrs. Baker became immortal and baked cookies every week for all eternity, how much sugar would she use?
Objectives: Solve exponential and logarithmic equations. Use exponential models to solve problems. If y = b x, then log b y = x. log x has a base of. ln is log with a base of. Properties of Logs: o log b M + log b N = o log b M log b N = o log b M a = To solve exponential equations: I. Isolate the power (base to the exponent). II. Write as a log. III. Solve for the variable if necessary. IV. Evaluate your answer in the calculator and round to the thousandths. To solve log equations: I. Condense logs into one log if necessary. II. Write in exponential form. III. Solve for x. IV. Check for extraneous solutions. (The argument of the log cannot be negative.) Examples: Solve each equation. Round your answers to the thousandths. 9. 13 x = 76 10. e 2x = 15 11. 6 14 x + 4 = 76 12. 3e x-5 = 7 13. 9 5x + 8 = 50 14. -5 2-7n 7 +5 = -66 15. -2 log 5 7x = 2 16. -6 log 3 (x 3) = -24 17. ln (4x 1) = 3 18. ln 2 + ln x = 1 19. log 8 2 + log 8 4x 2 = 1 20. log 2 (x + 1) log 2 (x 4) = 3
Objective: Use exponential models to solve problems. Exponential function: y = a b x where a = and b = If 0 < b < 1, then b represents and if b > 1, then b represents b = 1 + r for exponential growth, where r = growth % rate written in decimal form and b = 1 r for exponential decay where r = decay % rate written in decimal form. Half life: y = a ( 1 2 ) t n where a = initial amount, t = time, and n = half-life Examples: Solve each problem. 21. The population of Winnemucca, Nevada, can be modeled by P = 6191(1.04) t where t is the number of years since 1990. a) What was the population in 1990? b) By what percent did the population increase by each year? 22. You drink a beverage with 120 mg of caffeine. Each hour, the caffeine in your system decreases by about 12%. a) Write a model to represent the amount of caffeine in your system after x hours. b) How much caffeine is in your system after 3 hours? c) How long until you have 10mg of caffeine? Show work! 23. Selenium-83 has a half-life of 25 minutes. a) How much of a 18 mg sample is left after 2 hours? Show work! b) How many minutes would it take for a 10 mg sample to decay and have only 1.25 mg of it remain? Show work! 24. Iodine-131 has a half-life of 8 days. What fraction of the original sample would remain at the end of 32 days?
25. A medical institution requests 1 g of bismuth-214, which has a half life of 20 min. How many grams of bismuth-214 must be prepared if the shipping time is 2 h? 26. Carbon-14 dating works by measuring the amount of carbon-14 remaining in a fossil or artifact. Carbon-14 decays into nitrogen-14 at an exponential rate described by the formula: N(t) = N 0e -0.000124t where t is in years, N 0 is the original amount of carbon-14 and N(t) is the amount of carbon-14 remaining. a) An organism contained 100 milligrams of carbon-14 at the time of its death. How much carbon-14 remains after 100 years? b) What is the half-life of carbon-14? That is, how many years until an amount of carbon-14 is reduced by half? Show work! c) You re a collector of papyrus scrolls and learn that there s a set for sale an E-bay. The seller says that 70% of the original carbon-14 in the scrolls still remains, so she thinks it s about 3,500 years old. Using the formula N(t) = N 0e -0.000124t, what is your estimate of the age of the scrolls? Show work! Objective: Find the inverse of f(x). Steps for finding f 1 (x). I. Switch x and y. II. Solve for y. The inverse of a function is a function if the function passes the horizontal line test. Examples: Find the inverse of each function. 27. f(x) = 2x + 6 28. f(x) = x 3 5 29. f(x) = x 2 7 30. f(x) = 5 x 31. f(x) = 2 log x 32. f(x) = ln x 6 Determine whether the inverse of the function shown in each graph is a function. Explain. 33. 34.