Math topic中文

Concepts sequence series

Concepts of sequence and series for finite and infinite cases.

Practice questions

  1. 1Prove by the method of differences that ∑r=1N1r+r+1=N+1−1.\sum_{r=1}^{N} \frac{1}{\sqrt{r} + \sqrt{r+1}} = \sqrt{N+1} - 1.
  2. 2Consider Sn=21×2×3+22×3×4+23×4×5+…+2n(n+1)(n+2)S_{n}=\dfrac{2}{1\times2\times3}+\dfrac{2}{2\times3\times4}+\dfrac{2}{3\times4\times5}+\ldots+\dfrac{2}{n(n+1)(n+2)}…
  3. 3A sequence of numbers u1,u2,u3,…u_{1}, u_{2}, u_{3}, \ldots has sum SnS_{n} where Sn=∑r=1nurS_{n} = \sum_{r=1}^{n} u_{r}. It is given th…
  4. 4Using the formulae for sin⁡(A±B)\sin(A \pm B), prove that sin⁡(2r+1)θ−sin⁡(2r−1)θ=2cos⁡2rθsin⁡θ.\sin(2r + 1)\theta - \sin(2r - 1)\theta = 2 \cos 2r\theta \sin \theta .…
  5. 5已知定义：在数列{an}\{a_{n}\}中，若an2−an−12=p(n⩾2,n∈N∗,pa_{n}^{2}-a_{n-1}^{2}=p(n\geqslant 2,n∈N^{*},p为常数))，则称数列{an}\{a_{n}\}为等方差数列，下列判断： ①{(−1)n}①\{(-1)^{n}\}…
  6. 6Verify that 3(r+1)!−2r!−1(r−1)!=−r2+3r+1(r+1)!\dfrac{3}{(r+1)!} - \dfrac{2}{r!} - \dfrac{1}{(r-1)!} = -\dfrac{r^2 + 3r + 1}{(r+1)!}. Hence find ∑r=1n−r2−3r+1(r+1)!.\displaystyle\sum_{r=1}^{n} \dfrac{-r^2 - 3r + 1}{(r+1)!}. …
  7. 7Show that 7n−2−5n−1−2n=9n−4(n−2)(n−1)n\frac{7}{n-2} - \frac{5}{n-1} - \frac{2}{n} = \frac{9n-4}{(n-2)(n-1)n}. Hence find ∑n=3N9n−4(n−2)(n−1)n\sum_{n=3}^{N} \frac{9n-4}{(n-2)(n-1)n}…
  8. 8An arithmetic sequence has first term aa and common difference 0.50.5, where aa is an integer. Find the smallest value…
  9. 9Aenicillin, a type of bacteria, is grown in a petri dish. Its population doubles every six hours. Denoting the amount of…
  10. 10It is given that f(r)=(r−2)2r−1f(r)=(r-2)2^{r-1}. Show that f(r+1)−f(r)=2rarf(r+1)-f(r)=2^r a r, where aa is a constant to be determined. Hence f…
  11. 11Write 44r2+12r+5\dfrac{4}{4r^{2}+12r+5} in partial fractions. Find ∑r=2n44r2+12r+5\displaystyle\sum_{r=2}^{n} \dfrac{4}{4r^{2}+12r+5}. A sequ…
  12. 12A sequence u1,u2,u3,…u_1, u_2, u_3, \dots is such that u1=0u_1 = 0 and un+1=un+4+n−n2[(n+1)(n+2)]2, for all n∈Z+.u_{n+1} = u_n + \frac{4 + n - n^2}{[(n+1)(n+2)]^2}, \text{ for all } n \in \mathbb{Z}^+.…
  13. 13Dendrologists are specialised scientists who study trees and woody plants. Their work is diverse and can encompass vario…
  14. 14Given that ∑n=1Nn3=[N2(N+1)]2,\sum_{n=1}^{N} n^{3} = \left[\frac{N}{2}(N+1)\right]^{2}, Find 12−23+33−43+⋯−(2N)3,1^{2}-2^{3}+3^{3}-4^{3}+\dots-(2N)^{3},…
  15. 15Prove that 2n+1n2+2n+n2−1=n2+2n−n2−1.\displaystyle \frac{2n+1}{\sqrt{n^2+2n+\sqrt{n^2-1}}} = \sqrt{n^2+2n} - \sqrt{n^2-1}. Hence find ∑n=1N2n+1n2+2n+n2−1\displaystyle \sum_{n=1}^N \frac{2n+1}{\sqrt{n^2+2n}+\sqrt{n^2-1}}…
  16. 16In a computer simulation, each player uses an identical pail to pour water repeatedly into his assigned tank, which is i…
  17. 17Given that 1−2r=A(r+1)+Br1-2r = A(r+1) + Br, find the constants AA and BB. Use the method of differences to find ∑r=1n1−2r3r\displaystyle \sum_{r=1}^n \frac{1-2r}{3^r}…
  18. 18Verify that 3(r+1)!−2r!−1(r−1)!=−r2−3r+1(r+1)!\displaystyle \frac{3}{(r+1)!}-\frac{2}{r!}-\frac{1}{(r-1)!}=\displaystyle\frac{-r^2-3r+1}{(r+1)!}. Hence…
  19. 19Prove by induction that ∑r=1nr3=14n2(n+1)2.\sum_{r=1}^{n} r^{3} = \frac{1}{4} n^{2} (n+1)^{2}. The r th term of a series is given by ln⁡(2ar)\ln\left(2a^{r}\right)…
  20. 20Selena decides to borrow $200 000 from the bank to fund her home purchase on 1 January 2025. On the first day of each m…
  21. 21A sequence u1,u2,u3,…u_1, u_2, u_3, \dots is such that u1=e10u_1 = \dfrac{e}{10} and un−un−1=e2n−1u_n - u_{n-1} = \dfrac{e}{2^{n-1}}, for all n≥2n \ge 2…
  22. 22Two fair n-sided dice, each with faces numbered 1, 2, ..., n, (where n≥4n \ge 4) are tossed in an experiment. The n…
  23. 23Find ∑r=0n[(n+2)r+n]\sum_{r=0}^{n} [(n+2)r + n], giving your answer in terms of nn. [You may use the result ∑r=1nr3=14n2(n+1)2\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2…
  24. 24A financial institution, Future Investments Inc., has introduced a new investment scheme. The scheme pays a compound int…
  25. 25A data processing centre processes data (in suitable units) every day. Let unu_n, where nn is an integer, n≥1n \ge 1, re…
  26. 26An arithmetic series has positive first term aa and common difference dd. The first, fifth and second terms of the ari…
  27. 27设数列{an}\{a_{n}\}满足a1=1a_{1}\text{=}1，a2=2a_{2}\text{=}2，an+2=(1+cos⁡2nπ2)an+sin⁡2nπ2{a}_{n+2}=(1+{{\rm\cos}}^{2}\dfrac{nπ}{2}){a}_{n}+{{\rm\sin}}^{2}\dfrac{nπ}{2}…
  28. 28An arithmetic series has first term aa and common difference dd, where a>0a > 0 and d≠0d \ne 0. The first, sixth and nin…
  29. 29An arithmetic progression has first term aa and common difference dd, where aa and dd are non-zero. The first, third…
  30. 30Ryan has playing cards which he stacks into pyramids. He will begin by stacking up cards to form a pyramid with 1 level,…
  31. 31Given that ∑r=1n1r(r+1)=nn+1\sum_{r=1}^{n} \frac{1}{r(r+1)} = \frac{n}{n+1}. Show that ∑r=2n1r(r−1)=1−1n\sum_{r=2}^{n} \frac{1}{r(r-1)} = 1 - \frac{1}{n}…
  32. 32A sequence of negative real numbers x1,x2,x3,…x_1, x_2, x_3, \dots satisfies the relation xn+1=−1−2xn, for n≥1. x_{n+1} = -\sqrt{1-2x_n}, \text{ for } n \ge 1.…
  33. 33设数列{an}\{a_{n}\}的前nn项和为SnS_{n}，且 Sn=n2−4n+4S_{n}=n^{2}-4n+4． (1)求数列{an}\{a_{n}\}的通项公式； (2)设bn=an2nb_{n}= \dfrac {a_{n}}{2^{n}}…
  34. 34At a particular bank, a savings account pays an interest of 0.9% per month on the last day of each month. On 1 January 2…
  35. 35已知等差数列{an}\{a_{n}\}的公差d>0d>0，前nn项和为SnS_{n}，且满足a2a3=45a_{2}a_{3}=45，a1+a4=14.(1)a_{1}+a_{4}=14.(1)求数列{an}\{a_{n}\}的通项公式及前nn…
  36. 36It is given that ∑r=1n7r+4r(r+1)(r+2)=92−2n+1−5n+2.\sum_{r=1}^{n} \dfrac{7r+4}{r(r+1)(r+2)} = \dfrac{9}{2} - \dfrac{2}{n+1} - \dfrac{5}{n+2}. Find ∑r=72n7r−3r3−r\sum_{r=7}^{2n} \dfrac{7r-3}{r^3 - r}…
  37. 37A sequence u0,u1,u2,…u_0, u_1, u_2, \dots is such that u0=1u_0 = 1 and un=un−1+2n+1n(n+1),for all n∈N+.u_n = u_{n-1} + 2n + \frac{1}{n(n+1)}, \quad\text{for all } n \in \mathbb{N}^+.…
  38. 38It is given that f(n)=n5n−1\text{f}(n) = \frac{n}{5^{n-1}} where nn is a positive integer. By considering f(r)−f(r+1)\text{f}(r) - \text{f}(r+1)…
  39. 39Jeremy plans to invest $100 000 at the beginning of year 2024 into a current account which pays compound interest a…
  40. 40The diagrams below show a sequence of patterns formed by squares. Stage 0 is represented by a square of length 1. At eac…
  41. 41It is given that ∑r=1n2r(r+1)(r+2)=12−1(n+1)(n+2).\sum_{r=1}^n \dfrac{2}{r(r+1)(r+2)} = \dfrac{1}{2} - \dfrac{1}{(n+1)(n+2)}. State the sum to infini…
  42. 42Prove by induction that, for all positive integers nn and for x∈Rx \in \mathbb{R}, x≠0,x≠1x \ne 0, x \ne 1, 1x−1−1x−1x2−⋯−1xn=1xn(x−1).\dfrac{1}{x-1}-\dfrac{1}{x}-\dfrac{1}{x^2}-\dots-\dfrac{1}{x^n}=\dfrac{1}{x^n(x-1)}.…
  43. 43Given that ∑r=1n1r(r+1)=nn+1\displaystyle\sum_{r=1}^{n} \frac{1}{r(r+1)} = \frac{n}{n+1}. Show that ∑r=2n1r(r−1)=1−1n\displaystyle\sum_{r=2}^{n} \frac{1}{r(r-1)} = 1 - \frac{1}{n}…
  44. 44已知数列{an}\{a_{n}\}满足a1=a2=1a_{1}=a_{2}=1，an+2=an+1+an(n∈N∗).a_{n+2}=a_{n+1}+a_{n}(n\in N^{*}).记SnS_{n}为数列{1an}\{\dfrac{1}{a_{n}}\}的前nn项…
  45. 45已知数列{an}\{a_{n}\}满足a1=1a_{1}=1，a2=13a_{2}= \dfrac {1}{3}，若anan−1+2anan+1=3an−1an+1(n⩾2,n∈N+)a_{n}a_{n-1}+2a_{n}a_{n+1}=3a_{n-1}a_{n+1}(n\geqslant 2,n∈N^{+})…
  46. 46A sequence u1,u2,u3,…u_1, u_2, u_3, \dots is such that u1=−1u_1 = -1 and un+1=un+2(3−n)+au_{n+1} = u_n + 2(3^{-n}) + a, where n∈Z+n \in \mathbb{Z}^+…
  47. 47The sum of the first nn terms of a sequence, uru_r, is given by ∑r=1nur=1−n(n+1)!\displaystyle\sum_{r=1}^n u_r = 1 - \dfrac{n}{(n+1)!}…
  48. 48Given that ∑r=1nr2=n(n+1)(2n+1)6,\sum_{r=1}^{n} r^{2} = \frac{n(n+1)(2n+1)}{6}, find ∑r=1n(3r+1)2\sum_{r=1}^{n} (3r+1)^{2} in terms of n. Use your…
  49. 49A series is given by ∑r=1n2(4−3x)r\sum_{r=1}^{n} 2(4-3x)^r where xx is constant. Explain why this is a geometric series. Determine…
  50. 50It is given that ∑r=1n2r(r+1)(r+2)=12−1(n+1)(n+2)\displaystyle \sum_{r=1}^n \frac{2}{r(r+1)(r+2)} = \frac{1}{2} - \frac{1}{(n+1)(n+2)}. State the sum…
  51. 51已知数列{an}\{a_{n}\}满足a1=1ka_{1}=\dfrac{1}{k}，k⩾2k\geqslant 2，k∈N∗k\in N^{*}，[an][a_{n}]表示不超过ana_{n}的最大整数((如[1.6]=1)[1.6]=1)…
  52. 52Use the standard series for ln⁡(1+x)\ln(1+x) and ln⁡(1−x)\ln(1-x) to find the first three non-zero terms in the series for ln⁡(1+x1−x)\ln\left(\dfrac{1+x}{1-x}\right)…
  53. 53A financial institution, Future Investments Inc., has introduced a new investment scheme. The scheme pays a compound int…
  54. 54A drug 'Painomore' has a half-life of 4 hours in the bloodstream, where half-life of a drug is defined to be the time ta…
  55. 55Betty needs to decorate a wall of length 5 metres for a party. She attaches hooks, starting from the extreme left end of…
  56. 56At the beginning of May 2016, Sam borrowed $50 000 from a bank that charges him a special rate of 0.2% interest at the…
  57. 57A famous artist wants to create an artwork that uses squares only. On the first day, he draws the perimeter of a square…
  58. 58It is known that the nnth term of a sequence is given by un=ln⁡(nn+1)u_n = \ln\left(\frac{n}{n+1}\right), where n≥1n \ge 1. Determ…
  59. 59Prove by the method of mathematical induction that ∑r=2n2(r+3)(r+5)=1130−2n+9(n+4)(n+5) .\sum_{r=2}^{n}\frac{2}{(r+3)(r+5)}=\frac{11}{30}-\frac{2n+9}{(n+4)(n+5)}\,.…
  60. 60It is given that ∑r=1n7r+4r(r+1)(r+2)=92−2n+1−5n+2.\displaystyle\sum_{r=1}^{n} \frac{7r+4}{r(r+1)(r+2)} = \frac{9}{2} - \frac{2}{n+1} - \frac{5}{n+2}.…