Math topic中文

Recurrence relation

Sequence generated by the relation $u_{n+1} = \mathrm{f}(u_n)$ , including the use of a graphing calculator or a computer to generate the sequence.

Practice questions

1已知数列 $\{a_{n}\}$ 中， $a_{1}=\dfrac{7}{6}$ ， $a_{n}^{2}-a_{n}+1=a_{n+1}(n\in N^{*})$ ， $S_{n}$ 是 $\{\dfrac{1}{a_{n}}\}$ 的前…
2A sequence is such that $u_{0}=2$ and $u_{n}=u_{n-1}+n^{3}+\left(\dfrac{1}{2}\right)^{n}$ for $n\ge 1$ . (a) It is given…
3An arithmetic sequence has first term $a$ and common difference $0.5$ , where $a$ is an integer. Find the smallest value…
4Anand writes a computer programme to simulate a population of organisms in a controlled environment. It is assumed that…
5On the first day of last month, the temperature of a machine in a manufacturing plant was found to be $90^\circ\text{C}$ …
6Aenicillin, a type of bacteria, is grown in a petri dish. Its population doubles every six hours. Denoting the amount of…
7A sequence $u_1, u_2, u_3, \dots$ is such that $u_1 = \frac{1}{2}$ and $u_{n+1} = u_n - \frac{n^2 + n - 1}{(n+2)!}, \quad \text{for all } n \ge 1.$ …
8A team of cyclists plans to ride from Singapore to Penang and back over 10 days. The distance for the route taken from S…
9A sequence of negative real numbers $x_{1}, x_{2}, x_{3}, \dots$ satisfies the relation $x_{n+1} = -\sqrt{1-2x_{n}}, \text{ for } n\ge 1.$ …
10Jane decides to purchase an apartment. To finance her apartment, she intends to apply for a bank housing loan of $847 5…
11A sequence $u_1, u_2, u_3, \dots$ is such that $u_n = 2u_{n-1} - 15n^2 + 60n + A$ , where $A$ is a constant, and $n \ge 2$ …
12Write $\dfrac{4}{4r^{2}+12r+5}$ in partial fractions. Find $\displaystyle\sum_{r=2}^{n} \dfrac{4}{4r^{2}+12r+5}$ . A sequ…
13A sequence $u_1, u_2, u_3, \dots$ is such that $u_1 = 0$ and $u_{n+1} = u_n + \frac{4 + n - n^2}{[(n+1)(n+2)]^2}, \text{ for all } n \in \mathbb{Z}^+.$ …
14已知数列 $\{a_{n}\}$ 中， $a_{1}=1$ ， $a_{n}\boldsymbol{⋅}a_{n+1}=3^{n}$ ， $n\in N^{*}$ ，则下列说法正确的是 $(\quad)$ A. $a_{4}=9$ B…
15在数列 $\left\{{a}_{n}\right\}$ 中， $a_{1}=1$ ， $a_{n+1}-a_{n}=7-2n$ ，则数列 $\{a_{n}\}$ 中最大项的数值为___.
16A sequence follows the recurrence relation $U_{n+1}-U_n = 2\cos\left(\dfrac{(2n+1)x}{2}\right)\sin\dfrac{x}{2}$ , $U_1 = \sin x$ …
17已知正项数列 $\{a\}$ 满足 $n$ -- $an=anan-(n\geqslant ,n∈N+)$ ， $a$ ．设 $b= \dfrac {2^{n-1}}{a_{n}}$ ，求列 $\{bn\}$ 前项和 $Tn$ ．
18已知数列 $\{a_{n}\}$ 中， $a_{1}=1$ ， $a_{2}=2$ ，且 $a_{n+1}=4a_{n}-3a_{n-1}(n∈N^{*},n\geqslant 2)$ ． $($ Ⅰ $)$ 令 $b_{n}=a_{n+1}-a_{n}$ …
19Selena decides to borrow $200 000 from the bank to fund her home purchase on 1 January 2025. On the first day of each m…
20平面内有 $n$ 条直线，最多可将平面分成 $f(n)$ 个区域，则 $f(n)$ 的表达式为 $(\quad)$ A. $n+1\:\:\:\:\:\:\:\:\:\:$ B. $2n\:\:\:\:\:\:\:\:\:\:$ …
21A sequence of real numbers $x_1, x_2, x_3, \dots$ satisfies the recurrence relation $x_{n+1} = \frac{5x_n + 2}{2x_n + 3} \text{ for all } n \ge 1.$ …
22已知数列 $\left\{{{a}_{n}}\right\}$ 对任意的 $n\in{{\mathbf{N}}^{*}}$ ，都有 ${{a}_{n}}\in{{\mathbf{N}}^{*}}$ ，且 ${{a}_{n+1}}=\left\{\begin{array}{l}3{{a}_{n}}+1,{{a}_{n}}\mathrm{为奇数},\\ \dfrac{{{a}_{n}}}{2},{{a}_{n}}\mathrm{为偶数}.\text{ }\\\end{array}\right.$ …
23A hospital patient is given a pill containing 50 units of antibiotic at 12pm. By 1pm, the number of units of antibiotic…
24A data processing centre processes data (in suitable units) every day. Let $u_n$ , where $n$ is an integer, $n \ge 1$ , re…
25设数列 $\{a_{n}\}$ 满足 $a_{n+1}= \dfrac {4a_{n}-2}{a_{n}+1}(n∈N^{*})(1)$ 若 $a_{1}=3$ ， $b_{n}= \dfrac {2-a_{n}}{a_{n}-1}(n∈N^{*})$ …
26A sequence $u_1,u_2,u_3,\dots$ is such that $u_1 = \dfrac{3}{4}$ and $u_{n+1} = u_n + \dfrac{3}{4}\left(\dfrac{1}{2}\right)^{2n}$ …
27A hospital patient is given a pill containing 50 units of antibiotic at 12pm. By 1pm, the number of units of antibiotic…
28Sarah, a Singaporean Citizen, started a 4-year undergraduate Computing course at the National University of Singapore (N…
29设数列 $\{a_{n}\}$ 满足 $a_{1}\text{=}1$ ， $a_{2}\text{=}2$ ， ${a}_{n+2}=(1+{{\rm\cos}}^{2}\dfrac{nπ}{2}){a}_{n}+{{\rm\sin}}^{2}\dfrac{nπ}{2}$ …
30A sequence $u_0,\ u_1,\ u_2,\ \dots$ is such that $u_0 = 0$ and $u_n = u_{n-1} - 2^{-n}\left(n^2 - 4n + 2\right),\quad\text{for all } n \in \mathbb{Z}^+.$ …
31A sequence of negative real numbers $x_1, x_2, x_3, \dots$ satisfies the relation $x_{n+1} = -\sqrt{1-2x_n}, \text{ for } n \ge 1.$ …
32A sequence of positive numbers $x_1, x_2, x_3, \dots$ satisfies the recurrence relation $x_{n+1} = \sqrt{10 - 3x_n} \text{, for } n = 1, 2, 3, \dots$ …
33At 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…
34On 1st Jan 2023, Mr Kim borrows $\char36{}40000$ for his business from a bank. The bank provides two options for repaying the l…
35A sequence is such that $u_0 = 2$ and $u_n = u_{n-1} + n^3 + \left(\frac{1}{2}\right)^n$ for $n \ge 1$ . It is given that…
36A sequence $u_1, u_2, u_3, \dots$ is such that $u_1 = \frac{1}{3}$ and $u_{r+1} = u_r - \frac{1}{(2r-1)(2r+3)}, \quad \text{for all } r \ge 1.$ …
37A sequence $u_0, u_1, u_2, \dots$ is such that $u_0 = 1$ and $u_n = u_{n-1} + 2n + \frac{1}{n(n+1)}, \quad\text{for all } n \in \mathbb{N}^+.$ …
38Jeremy plans to invest $100 000 at the beginning of year 2024 into a current account which pays compound interest a…
39It is given that $\sum_{r=1}^n \dfrac{2}{r(r+1)(r+2)} = \dfrac{1}{2} - \dfrac{1}{(n+1)(n+2)}.$ State the sum to infini…
40已知数列 $\{a_{n}\}$ 满足 $a_{n}+a_{n+1}=2n$ ， $S_{n}$ 为其前 $n$ 项和，若 $a_{1}=a_{4}$ ，则 $S_{101}=(\quad)$ A. $4882$ B. $5100$ …
41Caleb is training for a vertical marathon, by running up the stairs of a building with 62 floors. He decides to train **…
42(1)三角形 $ABC$ 中， $AB=2\sqrt{3},BC=2,\angle C={{60}^{0}}$ ，则三角形 $ABC$ 的面积为___ (2)设等比数列 $\left\{ {{a}_{n}} \right\}$ 满足 $a{\,\!}_{1}+a{\,\!}_{3}=10$ …
43已知数列 $\{a_{n}\}$ 满足 $a_{1}=1$ ， $a_{2}= \dfrac {1}{3}$ ，若 $a_{n}a_{n-1}+2a_{n}a_{n+1}=3a_{n-1}a_{n+1}(n\geqslant 2,n∈N^{+})$ …
44A sequence of negative real numbers $x_1,x_2,x_3,\dots$ satisfies the relation $x_{n+1} = -\sqrt{1-2x_n},\text{ for } n\ge 1.$ …
45A company posted a video on a social media platform to advertise a new product. The video is uploaded at the start of $1^{\text{st}}$ …
46A sequence of numbers, $x_{n}$ , satisfy the relation $x_{n+1}=\tfrac{1}{3}\left(2x_{n}+\dfrac{1}{x_{n}^{2}}\right),\quad n\in\mathbb{N}^{+}\cup\{0\},\quad 0<x_{0}<1.$ …
47Caleb is training for a vertical marathon, by running up the stairs of a building with 62 floors. He decides to train th…
48A sequence $u_1, u_2, u_3, \dots$ satisfies the recurrence relation $u_n = \frac{n}{(n-1)^2}u_{n-1}$ , for $n \ge 2$ . Giv…
49A sequence of real numbers $x_{1}, x_{2}, x_{3}, \dots$ satisfies the recurrence relation $x_{n+1} = \dfrac{5x_{n} + 2}{2x_{n} + 3}\quad\text{for all } n \ge 1.$ …
50It is given that $\displaystyle \sum_{r=1}^n \frac{2}{r(r+1)(r+2)} = \frac{1}{2} - \frac{1}{(n+1)(n+2)}$ . State the sum…
51A first order recurrence relation for a sequence $u_0, u_1, u_2, \dots$ is an equation that relates the general term $u_n$ …
52A first order recurrence relation for a sequence $u_0, u_1, u_2, \dots$ is an equation that relates the general term $u_n$ …
53At the beginning of May 2016, Sam borrowed $50 000 from a bank that charges him a special rate of 0.2% interest at the…
54已知数列 $\{a_{n}\}$ 满足 $a_{1}\text{=}1$ ， $a_{n\text{+}1}\text{=}\dfrac{a_{n}}{2\text{+}3a_{n}}(n{\in}N^{\text{*}})$ ，则下列结论…
55A radio game show called the Inverted Tower of Ionah is broadcasted live everyday. Each day, a participant is invited to…
56If $u_0=\frac{1}{2}$ and $u_{n+1}=\dfrac{3u_n}{2u_n+1}$ for any non-negative integer $n$ , write down the values of $u_1$ …