Solutions to these questions are here but don't cheat- if the solution hasn't been written up it's because it wasn't requested so make sure to ask for it!
1) Find the sum to 15 terms of this series:
14 + 22 + 42 + ....... + [ (9+2n) + 3n] 2) The first term of a geometric sequence is a, where a <0. The common ratio of this sequence, r is less than -1. Which of the following graphs is most representative of the first 10 terms of the sequence?
3) You work for a company that pays you 1 cent for your first day of work, 2 cents on your second, 4 cents on your third, such that your wage doubles with each new day.
a) How much would you have earned in total if you worked for 31 days? b) How many days would you have to work to receive over $1,000,000 in total? 4) Mary plans to read a book in 7 days. Each day she reads 15 pages more than she read on the previous day. The book contains 1155 pages. Find the number of pages Mary must read on the first day to finish it in seven days.
5) Consider the following geometric series:
1 + 3x + 9x2 + .... a) For what values of x does this series have a limiting sum? b) The limiting sum of the series is 20. Find the value of x. 6) Tom wins $100,000 in a lottery and places it into an investment account which pays 9% interest per annum, compounded monthly. He plans to withdraw $M at the end of each year, immediately after the 12th interest payment and he wants the investment to last for 25 years. a)Write an expression for A1, the amount of money in the account immediately after his first withdrawal of $M. b) Show that A2 = 100000(1.0075)24 - M [1+(1.0075)]12 c) Find the value of M, to the nearest dollar, if the investment lasts for 25 years. 7) The third term of an arithmetic series is 32 and the sixth term is 17. i) Find the common difference. ii) Find the sum of the first 10 terms. 8) The first term of a geometric series is 16 and the fourth term is 1/4. i) Find the common ratio ii) Find the limiting sum of the series 9) the sum to n terms of an arithmetic series is 5n2 -11n for all positive integer values of n. What is the nth term of the sequence? 10) What numbered term will be the first term greater than 150 in the sequence: 7, 13, 19, 25 11) Insert 5 numbers between 64 and -14, so that the numbers form an arithmetic sequence. 12) A ball is dropped 25m onto a flat surface. It rebounds 20m and continues to bounce to 4/5 of its previous height. If it continues to bounce until it stops, through what total distance does the ball travel? 13) How many terms of the series 2 + 6 + 18 + 54 + ...... are needed to give a sum of 2186? 14) The fourth term of an arithmetic series is 27 and the seventh term is 12. What is the common difference? 15) A brand of rechargeable batteries provides power for 20 hours when first purchased fully charged. After its first recharge it only provides power for a further 18 hours. After its second recharge it only provides power for 16.2 hours. Each subsequent recharging results in the battery having 90% of its previous power available. How many hours could you expect to get out of the battery? 16) Find a number n, which when added to each of 2, 5 and 9 will give you a set of three numbers in geometric series. 17) The fourth term of a geometric series is 1 and the ninth term is 32. Find the sum of nine terms. 18) An investor wants to borrow $1,000,000 to purchase a block of units in Willoughby from BellaBank, which offers an interest rate of 6% p.a. reducible. The investor is to repay the loan in equal monthly instalments M, over 10 years. i) If An is the amount owing after n instalments, develop expressions for A1, A2 and An. Hence show the monthly instalment is given by:
ii) Calculate the value of the monthly instalment, M, to the nearest cent.
iii) Determine the amount still owing to BellaBank after 5 years, to the nearest cent. 19) What is the value of the following expression?
20) Find the sum of the first 30 terms of the arithmetic series 6 + 9 + 12 ......
21) An amateur athlete is training for a marathon and is planning her running schedule. She intends to increase the distance she runs each week leading up to the marathon. She does not want to injure herself, so each week (except for the last week), she plans to run 10% further than she ran the previous week. In her first week of training, she runs 40km. i) Write down an expression, in terms of n, for the distance she runs in her nth week of training. ii) There are 12 weeks of training before the marathon. In her final week of training, she plans to only run 20km. Find the total distance she has run in preparation for the marathon, to the nearest km. 22) A car dealership has a car for sale for a cash price of $20,000. It can also be bought on terms over 3 years. The first 6 months are interest free and after that interest is charged at a rate of 1% per month on that month's balance. Repayments are to be made in equal monthly instalments beginning at the end of the first month. A customer buys the car on these terms and agrees to monthly repayments of $M. Let $An be the amount owing at the end of the nth month. i) Find an expression for A6 ii) By first finding A7, show that: A8 = (20,000 - 6M)1.012 -M(1+1.01) iii) Find an expression for A36 iv) Assuming the car is paid off after 36 months, find the value of M to the nearest dollar. 23) Find the sum of the first 40 terms of the arithmetic series 40 + 36 + 32 + .... 24) The third and the seventh terms of a geometric series are 1.25 and 20 respectively. What is the first term> 25) The limiting sum of the series 1 - 2p +4p2 + ..... is 4/7. What is the value of p? 26) The sum of the first n terms of a certain arithmetic series is given by:
i) Calculate S1 and S2
ii) Find the first three terms of the sequence iii) Find an expression for the nth term of the sequence 27) An infinite geometric series has a first term of 4 and a common ratio of r and a limiting sum of 25r. Find the value(s) of r. 28) Find the first three terms of the arithmetic series for which T5 = 17 and T12 = 52. 29) Evaluate the following:
30) Ashleigh plans to deposit a sum of money into an account which guarantees to pay 1% interest each month on the balance of her account at the time. Immediately after each interest payment is made, Ashleigh intends to withdraw $500. She has no intention of ever adding to her initial deposit. Using M to signify the initial deposit and An to represent the value of the investment after n withdrawals,
i) write an expression for the value of her investment immediately after the first withdrawal. ii) Show that when she has made the third withdrawal, the balance of her account will be A3 = [M(1.01)3 - 500(1 + 1.01 + 1.012 ) ] iii) Write the expression for An iv) Ashleigh wants her deposit to be sufficient for her to be able to make withdrawals in this manner for 5 years. Show that her initial deposit needs to be $22,500 (to the nearest $100). 31) What is the limiting sum of the following geometric series if |x| <1: x2 -2x3 + 4x4 -8x5 + ..... 32) Find the sum of the first 25 terms of the series : 32 - 16 + 8 - 4 + 2 - ..... 33) Jessica invests $250 into an account at BellaBank. She invests the money at the beginning of each month for n years. Interest is to be paid at a rate of 6% p.a. compounded monthly. i) Show that the total value of her investment An at the end of n years is given by: An = $250(1.005 + 1.0052 + .... + 1.00512n ) ii) Find the value of the investment at the end of 7 years. iii) What single investment at the beginning of the 7 years would yield the same final value for Jessica? Assume interest is compounded monthly. 34) Jackson and Georgina take out a home loan of $460,000 over 30 years at 9% p.a. compounded monthly. The bank quote them a monthly repayment of $3701.26. M represents the monthly repayment and n represents the number of payments made. i) Show that the amount owing after 2 months is A2 = 460,000 (1.0075)2 - M(1+1.0075) ii) Find an expression for An iii) Jackson and Georgina decide to pay $5000 a month. Find the amount owing after 10 years. iv) Jackson and Georgina decide to start a family. They approach the bank and renegotiate their loan. After paying numerous fees and charges, they agree to pay off the remaining $160,000 at 7.5% p.a. compounded monthly. If their new repayment is $2500, find the time taken to repay the new loan to the nearest month. 35) Boxes are stacked in layers, where each layer contains one box less than the layer below. There are six boxes in the top layer, seven boxes in the next layer and so on. There are n layers altogether. What is the expression for the number of boxes in the bottom layer in terms of n? 36) Find the value of the following expression:
37) An infinite geometric series has a first term of 8 and a limiting sum of 12. What is the common ratio?
38) In 1910, John, an early settler of Mathsville, left a will in which he established a fund of $500 for its future citizens to spend on schools. His instructions were that his money was to be invested at 6% p.a. compounded yearly. i) If John's instructions were followed, how much would have been in the fund 100 years after it was established if the $500 was invested at the beginning of the year? iii) Suppose that at the beginning of each subsequent year after the establishment, a further $500 had been added to the fund and had also earned 6% interest compounded annually. Express the amount of money ($M) in the fund after 100 years as a geometric series and hence derive the value of M, correct to the nearest dollar. 39) A couple wishing to buy a holiday house require a loan of $180,000. The loan is to be repaid over 25 years in equal monthly instalments. The interest is 12% p.a. compounded monthly on the balance owing. i) Show that the size of each monthly repayment ($R) to the nearest dollar is $1896. ii) Find the total interest which will be paid. iii) Calculate the equivalent simple interest rate on the loan per annum. iv) How much is still owed on the property, to the nearest dollar, after 7 years? 40) Find the sum of the multiples of 4 between 23 and 157.
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