National J unior College Mathematics Department 2014 2013 2014 / H2 Maths / Revision Lecture (Arithmetic and Geometric Series)Page 1 of 2 National Junior College 2013 2014 H2 MathematicsRevision Lecture: Arithmetic and Geometric Series (Practice Set) Total Time: 1 hour [Total Marks: 32] [2012 TJC Prelim/P1/Q7] 1(a) A finite arithmetic progression has n terms and common difference d. The first term is 1 and the sum of the last 5 terms exceeds the sum of the first 4 terms by 193. (i)Show that 5nd 21d192 =0. [3] (ii)Given also that the 6th term of the progression is 16, find n.[2] (b) A sequence U is formed in which the nth term is given by entwhere ntis the nth term of an arithmetic progression with first term1t =1. (i)Show that U is a geometric progression. [2] (ii)Giventhatthesumtoinfinityofeven-numberedtermsofUis 8e63,findthe common ratio of U. [3] [2012 PJC Prelim/P1/Q9] 2A hamster farm has 500 hamsters to sell. The farmer sells khamsters at the end of every week, where k is a constant factor of 500 (e.g. 5, 100, etc.). The selling price of a hamster is $10 in the first week and it drops by 5% in each subsequent week. It is assumed that there is neither birth nor death of hamsters in the farm. (i)State the total number of weeks for the farmer to sell all his hamsters in terms of k . [1] (ii)Show that the total proceeds from selling all the hamsters is 500200 1 0.95kk . [3] (iii)Given that the cost of rearing a hamster is $0.50 a week, find the total cost incurred in rearing the hamsters, when the farmer has sold all his hamsters. [3] (iv)If all other costs are negligible, find the least value of k for the farmer to make a profit. [2] National J unior College Mathematics Department 2014 2013 2014 / H2 Maths / Revision Lecture (Arithmetic and Geometric Series)Page 2 of 2 [2012 GCE A Level/P2/Q4] 3On1J anuary2001MrsAput$100intoabankaccount,andonthefirstdayofeach subsequent month she put in $10 more than in the previous month. Thus on 1 February she put $110 into the account and on 1 March she put $120 into the account, and so on. The account pays no interest. (i)On what date did the value of Mrs As account first become greater than $5000?[5] On1J anuary2001MrBput$100intoasavingsaccount,andonthefirstdayofeach subsequentmonthheputanother$100intotheaccount.Theinterestratewas0.5%per month, so that on the last day of each month the amount in the account on that day was increased by 0.5%. (ii)Use the formula for the sum of a geometric progression to find an expression for the value of Mr Bs account on the last day of the nth month (where J anuary 2001 was the 1st month, February 2001 was the 2nd month, and so on). Hence find in which month the value of Mr Bs account first became greater than $5000.[5] (iii)MrBwantedthevalueofhisaccounttobe$5000on2December2003.What interest rate per month, applied from J anuary 2001, would achieve this?[3]