MISCELLANEOUS SEQUENCES & SERIES QUESTIONS

1 Created by T. Madas Created by T. Madas MISCELLANEOUS SEQUENCES & SERIES QUESTIONS Created by T. Madas Created by T. Madas Question 1 (**+) Show that 1212728430rrr= ++= . Detailed workings must be shown in this question. MP2-D, proof Created by T. Madas Created by T. Madas Question 2 (**+) Evaluate the following sum ()30132478rrr= . Detailed workings must support the answer. MP2-V, 715,822, 200 Created by T. Madas Created by T. Madas Question 3 (**+) Three numbers, A, B, C in that order, are in geometric progression with common ratio r.

2 Given further that A, 2B, C in that order are in arithmetic progression, determine the possible values of r. MP2-X,23r= Question 4 (**+) An arithmetic SERIES has common difference 2. The rd3, th6 and th10 terms of the arithmetic SERIES are the respective first three terms of a geometric SERIES . Determine in any order the first term of the arithmetic SERIES and the common ratio of the geometric SERIES . MP2-Z, 14a=, 43r= Created by T. Madas Created by T. Madas Question 5 (**+) Each of the terms of an arithmetic SERIES is added to the corresponding terms of a geometric SERIES , forming a new SERIES with first term 38 and second term 1316.

3 The common difference of the arithmetic SERIES is four times as large as the first term of the geometric SERIES . The common ratio of the geometric SERIES is twice as large as the first term of the arithmetic SERIES . Determine the possible values of the first term of the geometric SERIES . MP2-M, 7184 Created by T. Madas Created by T. Madas Question 6 (**) Solve the following equation +++++=. You may assume that the left hand side of the equation converges. MP2-U, 1x= Created by T. Madas Created by T. Madas Question 7 (**) Solve the following equation + + +=.

4 You may assume that the left hand side of the equation converges. MP2-X, 2134xx== Created by T. Madas Created by T. Madas Question 8 (**) Find in simplified form, in terms of n, the value of ()()21321nrrr= . SP-I, 3n Created by T. Madas Created by T. Madas Question 9 (**) The st1, rd3 and th11 term of an arithmetic progression are the first three terms of a geometric progression. It is further given that the sum of the first 13 terms of the arithmetic progression is 260. Find, in any order, the common ratio of the geometric progression and the first term and common difference of the arithmetic progression.

5 MP2-N, 4r= Created by T. Madas Created by T. Madas Question 10 (**) The nd2, rd3 and th9 term of an arithmetic progression are three consecutive terms of a geometric progression. Find the common ratio of the geometric progression. MP2-R, 6r= Created by T. Madas Created by T. Madas Question 11 (**+) It is given that 112nrrxn== and ()22211113nnrrrrxxnn== = . Determine, in terms of n, the value of ()211nrrx=+ . MP2-O, ()21118nrrxn=+= Created by T. Madas Created by T. Madas Question 12 (**+) It is given that ( )20110200rf r= = and ( )2021102800rf r= =.

6 Find the value of ( )2021rf r= . MP2-Q, ( )20218800rf r== Created by T. Madas Created by T. Madas Question 13 (**+) Solve the following equation ()2221 3 2x rxr ==+ . You may assume that the left hand side of the equation converges. SYN-P, 2x= Created by T. Madas Created by T. Madas Question 14 (**+) The sum of the first 2 terms of an arithmetic progression is 40. The sum of the first 4 terms of the same arithmetic progression is 130. a) Determine the sum of the first 5 terms of the arithmetic progression.

7 The sum of the first 2 terms of a geometric progression is 40. The sum of the first 4 terms of the same geometric progression is 130. b) Find the two possible values of the sum of the first 5 terms of the geometric progression. MP2-P, , 55211 or 275SS== Created by T. Madas Created by T. Madas Question 15 (**+) Consider the following 2 SEQUENCES . 10, 13, 16, 19, 22, .. and 6, 12, 24, 48, 96, .. The sum of the thnterm of the first sequence and the thnterm of the second sequence is denoted by nU. Show algebraically that ()13 1 2nnnUU+=++.

8 SYN-L, proof Created by T. Madas Created by T. Madas Question 16 (**+) Solve the following equation ()20sin2 tanrrxx == . You may assume that the left hand side of the equation converges. SYN-N, 1,0,1, 2, 3,..4xnn = = Created by T. Madas Created by T. Madas Question 17 (**) The product operator , is defined as [ ] uuuuu == . Solve the equation ()22122xrxr += = . You may assume that the left hand side of the equation converges. SPX-L, 1x= Created by T. Madas Created by T.

9 Madas Question 18 (**) It is given that 22111 32 58nnnrru++=+ = , where nu is the thn term of a sequence . Find a simplified expression for nu. SPX-C, 95nnnu= Created by T. Madas Created by T. Madas Question 19 (**) It is given that ()211324322nnrrunnn+== + + , where nu is the thn term of a sequence . Find a simplified expression for nu. SP-B, ()31265nnunn=+ + Created by T. Madas Created by T. Madas Question 20 (**) It is given that 11610 24nnnrru+== + , where nu is the thn term of a sequence .

10 Show clearly that 21nnnuAuBu++=+, where A and B are integers to be found. SP-P, 21812nnnuuu++= Created by T. Madas Created by T. Madas Question 21 (**) Find in exact simplified form an exact expression for the sum of the first n terms of the following SERIES +++++ SP-N, 111010 981nnSn+ = Created by T. Madas Created by T. Madas Question 22 (**) The first three terms of a geometric progression are the respective th7 term, th4 term, and nd2 term of an arithmetic progression. Determine the common ratio of the geometric progression.