Series 1 Arithmetic Progressions Exercise PDF

33. Write down the next two terms in each of the following sequences. 1. a) \ [ − 8,  − 5,  − 2, 1, 4, 7, ...]{.math.display}\ 2. b) \ [1, 4, 9, 16, 25, 36, ...]{.math.display}\ 3. c) \ [0.1, 0.01, 0.001, 0.0001, ...]{.math.display}\ 4. d) \ [\$\$\\frac{3}{4},\\ 1\\frac{1}{2},\...

33. Write down the next two terms in each of the following sequences. 1. a) \ [ − 8,  − 5,  − 2, 1, 4, 7, ...]{.math.display}\ 2. b) \ [1, 4, 9, 16, 25, 36, ...]{.math.display}\ 3. c) \ [0.1, 0.01, 0.001, 0.0001, ...]{.math.display}\ 4. d) \ [\$\$\\frac{3}{4},\\ 1\\frac{1}{2},\\ 2\\frac{1}{4},\\ 3,\\ 3\\frac{3}{4},\\ 4\\frac{1}{2},\\ \\ldots\$\$]{.math.display}\ 5. Write down the first term and common difference for each of the following A.P.s 6. a) \ [8 + 11 + 14 + 17 + ...]{.math.display}\ 7. b) \ [23 + 25 + 27 + 29 + ...]{.math.display}\ 8. c) \ [19 + 16 + 13 + 10 + ...]{.math.display}\ 9. d) \ [\$\$13\\frac{1}{2} + 15 + 16\\frac{1}{2} + 18 + \\ldots\$\$]{.math.display}\ 10. e) \ [ − 11.5 − 9 − 6.5 − 4 − ...]{.math.display}\ 11. f) \ [\$\$6\\frac{1}{4} + 6\\frac{3}{4} + 7\\frac{1}{4} + 7\\frac{3}{4} + \\ldots\$\$]{.math.display}\ 12. g) \ [ − 8 − 7 − 6 − 5 − ...]{.math.display}\ 13. h) \ [6 + 3 + 0 − 3 − ...]{.math.display}\ 14. Find the number of terms in each of the following A.P.s 15. a) \ [5 + 8 + 11 + 14 + ... + 59 + 62]{.math.display}\ 16. b) \ [1 + 6 + 11 + 16 + ... + 501 + 506]{.math.display}\ 17. c) \ [ − 193 − 189 − 185 − ... − 21 − 17]{.math.display}\ 18. d) \ [\$\$2\\frac{1}{4} + 2\\frac{17}{20} + 3\\frac{9}{20} + \\ldots + 20\\frac{1}{4} + 20\\frac{17}{20}\$\$]{.math.display}\ 19. Find the 18^th^ term of a series that has n^th^ term given by [2 + 3*n*]{.math.inline} 20. Find the 31^st^ term of a series that has n^th^ term given by [\$\\frac{1}{3}\\left( 10 + 2n \\right)\$]{.math.inline} 21. Find the 50^th^ term of a series that has n^th^ term given by [\$\\frac{1}{2}\\left( 32 - n \\right)\$]{.math.inline} 22. Find the 6^th^ and 7^th^ terms of a series that has n^th^ term given by [(−1)^*n*^(2*n*+1)]{.math.inline} 23. Find an expression for the n^th^ term for each of the following A.P.s and use your answer to write down the 100^th^ term of each series. 24. a) \ [5 + 8 + 11 + 14 + ...]{.math.display}\ 25. b) \ [5 + 2 − 1 − 4 − ...]{.math.display}\ 26. c) \ [\$\$12\\frac{1}{2} + 16 + 19\\frac{1}{2} + 23 + \\ldots\$\$]{.math.display}\ 27. For each of the following A.P.s, state which is the first term to exceed 1000. 28. a) \ [7 + 12 + 17 + 22 + 27 + ...]{.math.display}\ 29. b) \ [ − 24 − 21.5 − 19 − 16.5 − 14 − ...]{.math.display}\ 30. For each of the following A.P.s, state which is the first term to be negative. 31. a) \ [843 + 836 + 829 + 822 + ...]{.math.display}\ 32. b) \ [56.3 + 55.4 + 54.5 + 53.6 + ...]{.math.display}\ +-----------------------+-----------------------+-----------------------+ | 67. | State the values of | | | | [*a*]{.math.inline} | | | | and [*d*]{.math | | | |.inline} in each of | | | | the following A.P.s | | | | and find | | | | [*S*~*n*~]{.math | | | |.inline} as | | | | indicated. | | +=======================+=======================+=======================+ | 34. | a) | \ | | | | [2 + 6 + 10 + 14 + .. | | | |., *S*~12~]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 35. | b) | \ | | | | [10 + 8 + 6 + 4 + ... | | | | , *S*~15~]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 36. | c) | \ | | | | [4.5 + 6 + 7.5 + 9 +  | | | |..., *S*~19~]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 37. | d) | \ | | | | [15 + 13 + 11 + 9 + . | | | |.., *S*~16~]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 38. | e) | \ | | | | [7 + 3 − 1 − 5 − ..., | | | |  *S*~20~]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 39. | f) | \ | | | | [\$\$- 6\\frac{1}{2} | | | | - 5 - 3\\frac{1}{2} - | | | | 2 - | | | | \\ldots,S\_{12}\$\$]{ | | | |.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 40. | g) | \ | | | | [ − 9 − 7 − 5 − 3 − . | | | |.., *S*~16~]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 41. | Find the sum of each | | | | of the following | | | | A.P.s | | +-----------------------+-----------------------+-----------------------+ | 42. | a) | \ | | | | [2 + 4 + 6 + 8 + ...  | | | | + 146]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 43. | b) | \ | | | | [100 + 95 + 90 + 85 + | | | |  80 + ... − 20]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 44. | c) | \ | | | | [4 + 10 + 16 + 22 + 2 | | | | 8 + ... + 334]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 45. | d) | \ | | | | [\$\$5\\frac{1}{4} + | | | | 4\\frac{1}{2} + | | | | 3\\frac{3}{4} + | | | | \\ldots - | | | | 3\$\$]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 46. | In an A.P., | | | | [*u*~5~ = 8]{.math | | | |.inline} and | | | | [*u*~9~ = 14]{.math | | | |.inline}; find | | | | [*a*, *d*]{.math | | | |.inline} and | | | | [*S*~10~]{.math | | | |.inline} | | +-----------------------+-----------------------+-----------------------+ | 47. | In an A.P., | | | | [*u*~3~ = 7.5]{.math | | | |.inline} and | | | | [*u*~10~ = 11]{.math | | | |.inline}; find | | | | [*a*, *d*]{.math | | | |.inline} and | | | | [*S*~8~]{.math | | | |.inline} | | +-----------------------+-----------------------+-----------------------+ | 48. | In an A.P., | | | | [*u*~4~ = 15]{.math | | | |.inline} and | | | | [*u*~8~ = 7]{.math | | | |.inline}; find | | | | [*a*, *d*]{.math | | | |.inline} and | | | | [*S*~22~]{.math | | | |.inline} | | +-----------------------+-----------------------+-----------------------+ | 49. | In an A.P., | | | | [*u*~3~ =  − 4]{.math | | | |.inline} and | | | | [*u*~7~ = 8]{.math | | | |.inline}; find | | | | [*a*, *d*]{.math | | | |.inline} and | | | | [*S*~9~]{.math | | | |.inline} | | +-----------------------+-----------------------+-----------------------+ | 50. | In an A.P., | | | | [*u*~2~ =  − 12]{.mat | | | | h | | | |.inline} and | | | | [*S*~12~ = 18]{.math | | | |.inline}; find | | | | [*a*, *d*]{.math | | | |.inline} and | | | | [*u*~6~]{.math | | | |.inline} | | +-----------------------+-----------------------+-----------------------+ | 51. | In an A.P., | | | | [*u*~5~ =  − 0.5]{.ma | | | | th | | | |.inline} and | | | | [*S*~7~ = 21]{.math | | | |.inline}; find | | | | [*a*, *d*]{.math | | | |.inline} and | | | | [*u*~9~]{.math | | | |.inline} | | +-----------------------+-----------------------+-----------------------+ | 52. | In an A.P., | | | | [*u*~15~ = 7]{.math | | | |.inline} and | | | | [*S*~9~ = 18]{.math | | | |.inline}; find | | | | [*a*, *d*]{.math | | | |.inline} and | | | | [*u*~20~]{.math | | | |.inline} | | +-----------------------+-----------------------+-----------------------+ | 53. | The sum of the first | | | | 10 terms of an A.P. | | | | is 120 and the sum of | | | | the first 20 terms is | | | | 840. Find the sum of | | | | the first 30 terms. | | +-----------------------+-----------------------+-----------------------+ | 54. | An A.P. has a common | | | | difference | | | | [*d*]{.math.inline}. | | | | If the sum to 20 | | | | terms is 25 times the | | | | first term, find, in | | | | terms of [*d*]{.math | | | |.inline}, the sum to | | | | 30 terms. | | +-----------------------+-----------------------+-----------------------+ | 55. | In an A.P., | | | | [*a* =  − 23]{.math | | | |.inline} and | | | | [*d* = 2.5]{.math | | | |.inline}; find the | | | | least value of | | | | [*n*]{.math.inline} | | | | such that | | | | [*S*~*n*~ \> 0]{.math | | | |.inline} | | +-----------------------+-----------------------+-----------------------+ | 56. | In an A.P., | | | | [*a* =  − 61]{.math | | | |.inline} and | | | | [*d* = 4]{.math | | | |.inline}; find the | | | | least value of | | | | [*n*]{.math.inline} | | | | such that | | | | [*S*~*n*~ \> 0]{.math | | | |.inline} | | +-----------------------+-----------------------+-----------------------+ | 57. | An A.P. has first | | | | term 10 and common | | | | difference 0.25. Find | | | | the least number of | | | | terms the A.P. can | | | | have , given that the | | | | number of terms | | | | exceeds 300. | | +-----------------------+-----------------------+-----------------------+ | 58. | An A.P. has first | | | | term [ − 5]{.math | | | |.inline} and common | | | | difference 1.5. Find | | | | the greatest number | | | | of terms the A.P. can | | | | have, given that the | | | | sum of the terms does | | | | not exceed 450. | | +-----------------------+-----------------------+-----------------------+ | 59. | The sum to | | | | [*n*]{.math.inline} | | | | terms of a particular | | | | series is given by | | | | [*S*~*n*~ = 17*n* − 3 | | | | *n*^2^]{.math | | | |.inline}. | | +-----------------------+-----------------------+-----------------------+ | 60. | a) | Find an expression | | | | for the sum to | | | | [(*n*−1)]{.math | | | |.inline} terms. | +-----------------------+-----------------------+-----------------------+ | 61. | b) | Find an expression | | | | for the | | | | [*n*^th^]{.math | | | |.inline} term of the | | | | series. | +-----------------------+-----------------------+-----------------------+ | 62. | c) | Show that the series | | | | is an A.P. and find | | | | the first term and | | | | the common | | | | difference. | +-----------------------+-----------------------+-----------------------+ | 63. | Three consecutive | | | | terms of an A.P. have | | | | a sum of 36 and a | | | | product of 1428. Find | | | | the three terms. | | +-----------------------+-----------------------+-----------------------+ | 64. | A particular A.P. has | | | | a positive common | | | | difference and is | | | | such that for any | | | | three adjacent terms, | | | | three times the sum | | | | of their squares | | | | exceeds the square of | | | | their sum by 37.5. | | | | Find the common | | | | difference. | | +-----------------------+-----------------------+-----------------------+ | 65. | Find the common | | | | difference, the n^th^ | | | | term and the sum to | | | | [*n*]{.math.inline} | | | | terms of the | | | | following A.P. | | | | | | | | \ | | | | [ln 3 + ln 3^2^ + ln  | | | | 3^3^ + ln 3^4^ + ...] | | | | {.math | | | |.display}\ | | +-----------------------+-----------------------+-----------------------+ | 66. | Find expressions for | | | | the n^th^ term and | | | | the sum to | | | | [*n*]{.math.inline} | | | | terms of the A.P. | | | | | | | | \ | | | | [ln (*ab*) + ln (*ab* | | | | ^2^) + ln (*ab*^3^) + | | | |  ln (*ab*^4^) + ...]{ | | | |.math | | | |.display}\ | | +-----------------------+-----------------------+-----------------------+