RVHS H2 Math APGP & Series Qns

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Sequences & Series 1 Sequences and Series 1. AJC/2013/I/6 (a) The first term of an increasing arithmetic progression is 1. Sn is the sum of the first n terms of the arithmetic progression and S5, S10 and S20 form a geometric progression. Show that the common difference of the arithmetic progression is 2. [3] Without the use of a graphing calculator, find the least value of n such that 1 2 1100 n n S S + − . [3] (b) Mr Wee borrowed $3400 from an unlicensed money lender on 1st June 2013. The amount he owes the lender at the beginning of each month is twice the outstanding amount at the end of the previous month. From the month of August 2013, Mr Wee decided to repay $7000 i n the middle of each month. Find, in terms of n, the outstanding amount owed at the end of the nth month. Hence find the earliest month when he has fully repaid his loan. [5] 2. ACJC Prelim/2020/01/Q8 A hollow circular cylinder of diameter 42 mm is wrapped with n layers of thin film. The first layer of film is in contact with the outer surface of the cylinder and it has length 42  mm. The second layer is in contact with the first layer and has length (42 + 2 x) mm, where x mm is the thickness of the thin film. There is no gap between any two layers. The nth layer has length 124 mm. (i) Show that the thickness of the film satisfies the equation x(n – 1) = 41. [2] (ii) Hence find x, given that the total sum of the lengths of the n layers of film is 16766  mm. [3] Another roll of film which has length 52580 mm is to be cut into n pieces of increasing lengths, where the lengths form a geometric progression with common ratio r. It is given that the sum of the lengths of the first ten pieces of film is k times the sum of the lengths of the first five pieces of film, where k is an unknown constant. (iii) Show that 5 1kr=+ . [2] (iv) Given that k = 33 and that the first piece of film is of length 50 mm, find the largest possible value of n. [3]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Sequences & Series 2 3. TJC/2014/I/7 A convergent geometric sequence of positive terms, G has first term a and common ratio r. Write down, in terms of a and r, an expression for the nth odd-numbered term of G. [1] If the sum of first n odd-numbered terms of G is equal to the sum of all terms of G after the nth odd- numbered term, show that 2 2 12 1 0nnrr −+ − = . (i) Hence find the value of r when 5n= . [3] (ii) In another sequence H, each term is the reciprocal of the corresponding term of G. If the nth term of G and H is denoted by nu and nv respectively, show that a new sequence whose nth term is ln n n u v    , is an arithmetic progression. [4] 4. CJC/2013/II/1 (i) A certain machine A is pumping a liquid into an empty container of total volume 850 m3. The first pumping action fills the container with a volume of 4 m 3. Each subsequent pumping action fills the container with 0.5 m 3 more than the previous pumping action. Pumping continues until the container is completely filled with the liquid. Find the volume of liquid that overflows from the container at the final pumping action. [3] (ii) Suppose that, after the 50th pumping action by machine A, a different machine B is used instead to fill the remaining volume of the container. Using machine B, the first pumping action fills the container with a volume of 5 m3. Each subsequent pumping action fills the container with the amount filled in the previous pumping action. (a) Find the total volume of the container filled after the 10 th pumping action by machine B. [3] (b) Explain whether the container would be completely filled eventually. [2] 5. VJC/I/13 The sum of the first 5 terms of an arithmetic progression, where the common difference is non-zero, is 105. It is further given that the first, third and fourth terms of the arithmetic progression are the first 3 terms of a geometric progression. Show that the common difference of the arithmetic progression is 21 2− . [3] Deduce that the geometric progression is convergent. [2] Let S be the sum to infinity of the geometric progression, and An be the sum to n terms of the arithmetic progression. Find the least value of n such that the sum of 7S and An is not positive. [4] 5 6

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Sequences & Series 3 6. EJC Prelim/2020/02/Q5 On 1 Jan 2019, Emma takes up a study loan of $30000 from a bank that offers two repayment plans. Plan A charges a monthly instalment of $600 at the beginning of every month and increases by $50 for every subsequent month. There is no interest charged; inst ead, the bank charges a one -time administrative fee of $2500 which is to be added to the study loan. (i) (a) Show that, after the nth payment, she would have paid a total amount of $ ( ) 225 575nn+ .[2] (b) How many payments will it take for Emma to fully repay her loan? [2] Plan B charges a fixed monthly instalment of $x which is to be paid at the beginning of each month, starting from 1 Jan 2019. There is no interest charged for the first month only. Thereafter, interest is charged at the end of each month on the outstanding amount at 1% per month, (i.e. interest will be first charged on 28 February 2019.) (ii) (a) Show that the amount she owes after the fourth payment is ( ) ( ) ( ) ( ) 22 1.01 30000 1.01 2 1.01 x x x− − − . [2] (b) Use the formula for the sum of a geometric progression to find an expression, in terms of n and x, the amount she owes after the nth payment, for 2n . [2] (c) If Emma intends to repay the loan fully after 32 payments, find the least value of x. [2] (iii) Using the value of x in part (ii)(c), determine which plan will be cheaper for Emma. [1] 7. NJC/2010/I/8 (a) The sum, nS , of the first n terms of an arithmetic progression is given by 2 2nS n n=− . Write down the expression for 1nnSS −− . Hence, find the value of the common difference. (b) A metal screw of length L (measured in millimetre) is driven into a concrete wall by an electrical screwdriver, such that its distance driven into the wall is proportional to the angle turned by the screwdriver. Due to some reasons, every subsequent turn by this electrical screwdriver can only achieve an 80% of the angle turned previously. (i) Given that the initial angle turned by the screwdriver is  radians, write down the expressions for the first 3 distances driven into the concrete wall, leaving your answers in terms of  and k, where k is the constant of proportionality. (ii) Find the total distance driven into the concrete walls after n turns, leaving your answer in the form ( )1 nak b − , where a and b are constants to be determined. (iii) Given that 2k = and assuming that the total distance driven could never exceed the length of the screw, find the minimum length of the metal screw required, giving your answer in terms of  . L

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Sequences & Series 4 8. ACJC/2010/I/10 (a) The first 2 terms of a geometric progression are a and b ( b < a ). If the sum of the first n terms is equal to twice the sum to infinity of the remaining terms, prove that