Chapter Test 907 In Problems 22–25, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 22. + + + + 3 1 1 3 1 9 23. − + − + 2 1 1 2 1 4 24. + + + 1 2 3 4 9 8 25. ∑ ( ) = ∞ − 4 1 2 k k 1 1 In Problems 30–32, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers. 30. ( ) + + + + = + n n n 3 6 9 3 3 2 1 31. + + + + ⋅ = − − 2 6 18 2 3 3 1 n n 1 32. ( ) ( ) + + + + − = − − n n n n 1 4 7 3 2 1 2 6 3 1 2 2 2 2 2 33. Evaluate: ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 5 2 In Problems 34 and 35, expand each expression using the Binomial Theorem. 34. ( ) +x 2 5 35. ( ) −x3 4 4 36. Find the coefficient of x7 in the expansion of ( ) +x 2 . 9 37. Find the coefficient of x2 in the expansion of ( ) +x2 1 . 7 38. Constructing a Brick Staircase A brick staircase has a total of 25 steps. The bottom step requires 80 bricks. Each step thereafter requires three fewer bricks than the prior step. (a) How many bricks are required for the top step? (b) How many bricks are required to build the staircase? 39. Creating a Floor Design A mosaic tile floor is designed in the shape of a trapezoid 30 feet wide at the base and 15 feet wide at the top. The tiles, 12 inches by 12 inches, are to be placed so that each successive row contains one fewer tile than the row below. How many tiles will be required? 40. Bouncing Balls A ball is dropped from a height of 20 feet. Each time it strikes the ground, it bounces up to threequarters of the height of the previous bounce. (a) What height will the ball bounce up to after it strikes the ground for the 3rd time? (b) How high will it bounce after it strikes the ground for the nth time? (c) How many times does the ball need to strike the ground before its bounce is less than 6 inches? (d) What total distance does the ball travel before it stops bouncing? 41. Retirement Planning Chris gets paid once a month and contributes $350 each pay period into his 401(k). If Chris plans on retiring in 20 years, what will be the value of his 401(k) if the per annum rate of return of the 401(k) is 6.5% compounded monthly? 42. Salary Increases Your friend has just been hired at an annual salary of $50,000. If she expects to receive annual increases of 4%, what will be her salary as she begins her 5th year? 43. Saving for a Computer Patrice wants to buy a new computer gaming system for $2200. On March 1, Patrice deposits $700 in a money market account that pays 1.5% per annum compounded monthly. Starting on April 1, he plans to deposit $125 every month at the beginning of each month into the money market until he has enough saved for the computer. (a) Write a recursive sequence that explains how much is in Patrice’s account after n months. (b) After how many months will Patrice be able to purchase the computer gaming system? In Problems 26 and 27, determine whether each sequence converges or diverges. If it converges, find its limit. 26. { } { } = + s n n 7 2 n 2 2 27. { } { } = + a n 3 1 n n 28. Define infinite series. 29. Define what it means for a sequence to converge. The Chapter Test Prep Videos include step-by-step solutions to all chapter test exercises. These videos are available in MyLab™ Math. In Problems 1 and 2, list the first five terms of each sequence. 1. { } { } = − + s n n 1 8 n 2 2. = = + − a a a 4, 3 2 n n 1 1 In Problems 3 and 4, expand each sum. Evaluate each sum. 3. ∑ ( ) ( ) − + = + k k 1 1 k k 1 3 1 2 4. ∑ ( ) − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = k 2 3 k k 1 4 5. Write the following sum using summation notation. − + − + + 2 5 3 6 4 7 11 14 Chapter Test
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