Math topic中文

Sequence as function

Sequence as a function y=f(n)y = \mathrm{f}(n) where nn is a positive integer.

Practice questions

  1. 1The sum of the first nn terms of a series is given by Sn=3n(n+2)S_n = 3n(n + 2). Show that this series follows an arithmetic pr…
  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, \dots has a sum SnS_n where Sn=∑r=1nurS_n = \sum_{r=1}^{n} u_r. It is given that Sn=An2(n+1)−BnS_n = An^2(n + 1) - Bn…
  4. 4Given f(r)=r(r−1)!f(r)=\dfrac{r}{(r-1)!}, show that f(r−1)−f(r)=r2−3r+1(r−1)!f(r-1)-f(r)=\dfrac{r^2-3r+1}{(r-1)!}. Hence find ∑r=2nr2−3r+1(r−1)!\displaystyle\sum_{r=2}^n \dfrac{r^2-3r+1}{(r-1)!}…
  5. 5Show 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}…
  6. 6Anand writes a computer programme to simulate a population of organisms in a controlled environment. It is assumed that…
  7. 7On the first day of last month, the temperature of a machine in a manufacturing plant was found to be 90∘C90^\circ\text{C}…
  8. 8The first four terms of a sequence are given by v1=−9v_1 = -9, v2=7v_2 = 7, v3=47v_3 = 47 and v4=141v_4 = 141. It is given that vnv_n…
  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. 11A team of cyclists plans to ride from Singapore to Penang and back over 10 days. The distance for the route taken from S…
  12. 12Jane decides to purchase an apartment. To finance her apartment, she intends to apply for a bank housing loan of $847 5…
  13. 13Dendrologists are specialised scientists who study trees and woody plants. Their work is diverse and can encompass vario…
  14. 14Prove 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}}…
  15. 15Wendie attends training sessions in preparation for a running race. In the first session, she runs a distance of 1000 me…
  16. 16A sequence TnT_n is given by Tn=extln(na)+bn2+cT_n = ext{ln}(n^a) + bn^2 + c for nextoldsymbolmackslashinextoldsymbolmackslashZ+n ext{ } oldsymbol{ m ackslash in} ext{ } oldsymbol{ m ackslash Z}^+…
  17. 17Selena decides to borrow $200 000 from the bank to fund her home purchase on 1 January 2025. On the first day of each m…
  18. 18A 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…
  19. 19Cadence, the pedal speed in cycling, is measured in pedal stroke revolutions per minute (RPM). For example, a cadence of…
  20. 20The graph of y=x+40y = \sqrt{x+40} is shown in the diagram below. By considering the shaded rectangle, show that n+40<∫nn+1x+40 dx.\sqrt{n+40} < \int_n^{n+1} \sqrt{x+40}\, dx.…
  21. 21Sarah, a Singaporean Citizen, started a 4-year undergraduate Computing course at the National University of Singapore (N…
  22. 22An arithmetic series has first term aa and common difference dd, where a>0a > 0 and d≠0d \ne 0. The first, sixth and nin…
  23. 23It is given that f(r)=1−3(r+2)!\mathrm{f}(r) = 1 - \frac{3}{(r+2)!}, where rr is a non-negative integer. Show that f(r)−f(r−1)=a(r+1)(r+2)!\mathrm{f}(r) - \mathrm{f}(r-1) = \frac{a(r+1)}{(r+2)!}…
  24. 24The first nn terms of a series are given by log⁡a3+log⁡a27+log⁡a243+…+log⁡a32n−1\log_a 3 + \log_a 27 + \log_a 243 + \ldots + \log_a 3^{2^{n-1}}, where aa…
  25. 25Given f(r)=r(r−1)!\mathrm{f}(r) = \frac{r}{(r-1)!}, show that f(r−1)−f(r)=r2−3r+1(r−1)!\mathrm{f}(r-1) - \mathrm{f}(r) = \frac{r^2 - 3r + 1}{(r-1)!}.
  26. 26A geometric series has common ratio rr, and an arithmetic series has first term aa and common difference dd, where aa…
  27. 27等差数列{an}\{a_{n}\}的前nn项和为SnS_{n}，已知a1=13a_{1}=13，S3=S11S_{3}=S_{11}，当SnS_{n}最大时，nn的值是()() A. 55 B. 66 C. 77…
  28. 28At 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…
  29. 29It 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)…
  30. 30Jeremy plans to invest $100 000 at the beginning of year 2024 into a current account which pays compound interest a…
  31. 31The diagrams below show a sequence of patterns formed by squares. Stage 0 is represented by a square of length 1. At eac…
  32. 32已知SnS_{n}是数列{n2n−1}\{\dfrac{n}{{{2}^{n-1}}}\}的前nn项和，若不等式∣λ+1∣<Sn+n2n−1|\lambda +1| < {{S}_{n}}+\dfrac{n}{{{2}^{n-1}}}对一切n∈N∗n∈N^{*}…
  33. 33NYJC Promo 9758/2020/Q2 The first four terms of a sequence unu_n are given by u1=32.1u_1 = 32.1, u2=17u_2 = 17, u3=0.7u_3 = 0.7 and…
  34. 34The 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)!}…
  35. 35已知数列{an}\{a_{n}\}，an=1n(n+2)(n∈N+)a_{n}= \dfrac {1}{n(n+2)}(n∈N_{+})，那么1120 \dfrac {1}{120}是这个数列的第 ___ 项.
  36. 36A financial institution, Future Investments Inc., has introduced a new investment scheme. The scheme pays a compound int…
  37. 37Betty needs to decorate a wall of length 5 metres for a party. She attaches hooks, starting from the extreme left end of…
  38. 38At the beginning of May 2016, Sam borrowed $50 000 from a bank that charges him a special rate of 0.2% interest at the…
  39. 39A famous artist wants to create an artwork that uses squares only. On the first day, he draws the perimeter of a square…
  40. 40A series is given by ∑r=1n2(4−3x)r\displaystyle\sum_{r=1}^{n}2(4-3x)^{r} where xx is constant. Explain why this is a geometric ser…
  41. 41A radio game show called the Inverted Tower of Ionah is broadcasted live everyday. Each day, a participant is invited to…
  42. 42A complex number zz is given by z=reiπnz = r \mathrm{e}^{\mathrm{i}\frac{\pi}{n}}, where 0<r<10 < r < 1 and nn is a positive i…
  43. 43已知数列{xn}\{x_{n}\}和{yn}\{y_{n}\}的通项公式分别为xn=anx_{n}=a^{n}和yn=(a+1)n+b,n∈N+y_{n}=(a+1)n+b,n∈N^{+}． (1)当a=3a=3，b=5b=5时， ①①试问：x2x_{2}…