Math topic中文

Formula nth term

Sequence given by a formula for the $n$ th term.

Practice questions

1Prove by the method of differences that $\sum_{r=1}^{N} \frac{1}{\sqrt{r} + \sqrt{r+1}} = \sqrt{N+1} - 1.$
2A sequence of numbers $u_{1}, u_{2}, u_{3}, \ldots$ has sum $S_{n}$ where $S_{n} = \sum_{r=1}^{n} u_{r}$ . It is given th…
3A sequence of numbers $u_1, u_2, u_3, \dots$ has a sum $S_n$ where $S_n = \sum_{r=1}^{n} u_r$ . It is given that $S_n = An^2(n + 1) - Bn$ …
4By writing $\frac{3}{(3r+1)(3r+4)}$ in partial fractions, find $\sum_{r=1}^{n} \frac{3}{(3r+1)(3r+4)}$ in terms of $n$ .…
5A 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…
6数列 ${-}1{,}5{,}{-}9{,}13{,}{…}$ 的一个通项公式是 $a_{n}{=}$ ___ ．
7An arithmetic sequence has first term $a$ and common difference $0.5$ , where $a$ is an integer. Find the smallest value…
8The sum of the first $n$ terms of a sequence, $S_n$ , is given by $S_n = an^3 + bn^2 + cn + d$ , where $a$ , $b$ , $c$ and $d$ …
9The first four terms of a sequence are given by $v_1 = -9$ , $v_2 = 7$ , $v_3 = 47$ and $v_4 = 141$ . It is given that $v_n$ …
10数列 $\{a_{n}\}$ 中， $S_{n}=4-a_{n}- \dfrac {1}{2^{n-2}}$ ． $($ Ⅰ $)$ 求 $a_{1}$ ， $a_{2}$ ， $a_{3}$ ， $a_{4}$ ； $($ Ⅱ $)$ …
11A sequence $u_1, u_2, u_3, \dots$ is such that $u_1 = \frac{1}{2}$ and $u_{n+1} = u_n - \frac{n^2 + n - 1}{(n+2)!}, \quad \text{for all } n \ge 1.$ …
12A 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 $n \ge 2$ …
13已知等差数列 $\{a_{n}\}$ 的通项公式 $a_{n}= \dfrac {64-4n}{5}$ ，设 $A_{n}=|a_{n}+a_{n+1}+…+a_{n+12}|(n∈N^{*})$ ，当 $A_{n}$ 取得最小值时， $n$ …
14A sequence $u_1, u_2, u_3, \dots$ is such that $u_1 = 0$ and $u_{n+1} = u_n + \frac{4 + n - n^2}{[(n+1)(n+2)]^2}, \text{ for all } n \in \mathbb{Z}^+.$ …
15Dendrologists are specialised scientists who study trees and woody plants. Their work is diverse and can encompass vario…
16Given that $\sum_{n=1}^{N} n^{3} = \left[\frac{N}{2}(N+1)\right]^{2},$ Find $1^{2}-2^{3}+3^{3}-4^{3}+\dots-(2N)^{3},$ …
17Prove that $\displaystyle \frac{2n+1}{\sqrt{n^2+2n+\sqrt{n^2-1}}} = \sqrt{n^2+2n} - \sqrt{n^2-1}.$ Hence find $\displaystyle \sum_{n=1}^N \frac{2n+1}{\sqrt{n^2+2n}+\sqrt{n^2-1}}$ …
18Prove by induction that $\sum_{r=1}^{n} r^{3} = \frac{1}{4} n^{2} (n+1)^{2}.$ The r th term of a series is given by $\ln\left(2a^{r}\right)$ …
19已知数列 $\{a_{n}\}$ 的前 $n$ 项为和 $S_{n}$ ，点 $(n, \dfrac {S_{n}}{n})$ 在直线 $y= \dfrac {1}{2}x+ \dfrac {11}{2}$ 上.数列 $\{b_{n}\}$ …
20A sequence $T_n$ is given by $T_n = ext{ln}(n^a) + bn^2 + c$ for $n ext{ } oldsymbol{ m ackslash in} ext{ } oldsymbol{ m ackslash Z}^+$ …
21A sequence follows the recurrence relation $U_{n+1}-U_n = 2\cos\left(\dfrac{(2n+1)x}{2}\right)\sin\dfrac{x}{2}$ , $U_1 = \sin x$ …
22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 $b_{n}=a_{n+1}-a_{n}$ …
23Selena decides to borrow $200 000 from the bank to fund her home purchase on 1 January 2025. On the first day of each m…
24A sequence $u_1, u_2, u_3, \dots$ is such that $u_1 = \dfrac{e}{10}$ and $u_n - u_{n-1} = \dfrac{e}{2^{n-1}}$ , for all $n \ge 2$ …
25The reproduction number, $R_0$ , of a disease is a mathematical term that indicates how contagious an infectious disease…
26Use the standard series for $\ln(1+x)$ and $\ln(1-x)$ to find the first three non-zero terms in the series for $\displaystyle\ln\left(\frac{1+x}{1-x}\right).$ …
27A bag contains 10 red and 15 green beads which are indistinguishable apart from colour. Beads are drawn from the bag sin…
28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. $2n\:\:\:\:\:\:\:\:\:\:$ …
29已知数列 $\{a_{n}\}$ 中， $a_{1}=2$ ， $a_{2}=3$ ，其前 $n$ 项和 $S_{n}$ 满足 $S_{n+1}+S_{n-1}=2S_{n}+1$ ，其中 $n\geqslant 2$ ， $n∈N^{*}$ …
30Show that $\dfrac{1}{\sqrt{a}+\sqrt{b}}=\dfrac{\sqrt{a}-\sqrt{b}}{a-b}$ . Find $\displaystyle\sum_{r=1}^{2500}\dfrac{1}{\sqrt{4r-3}+\sqrt{4r+1}}$ …
31Find $\sum_{r=0}^{n} [(n+2)r + n]$ , giving your answer in terms of $n$ . [You may use the result $\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2$ …
32A data processing centre processes data (in suitable units) every day. Let $u_n$ , where $n$ is an integer, $n \ge 1$ , re…
33A sequence $u_1,u_2,u_3,\dots$ is such that $u_1 = \dfrac{3}{4}$ and $u_{n+1} = u_n + \dfrac{3}{4}\left(\dfrac{1}{2}\right)^{2n}$ …
34Sarah, a Singaporean Citizen, started a 4-year undergraduate Computing course at the National University of Singapore (N…
35A sequence $u_0,\ u_1,\ u_2,\ \dots$ is such that $u_0 = 0$ and $u_n = u_{n-1} - 2^{-n}\left(n^2 - 4n + 2\right),\quad\text{for all } n \in \mathbb{Z}^+.$ …
36The sum, $S_{n}$ , of the first $n$ terms of a sequence $u_{1}, u_{2}, u_{3}, \dots$ is given by $S_{n}=a(3^{n})+bn+c$ , w…
37已知等差数列 $\{a_{n}\}$ 的首项 $a_{1}=-2$ ，等比数列 $\{b_{n}\}$ 的公比为 $q$ ，且 $a_{2}=b_{1}$ ， $a_{3}=b_{2}+1$ ， $a_{1}b_{2}+5b_{2}=b_{3}$ …
38数列 $\{a_{n}\}$ 的通项公式是 $a_{n}=21+4n-n^{2}$ ，这个数列从第 ___ 项起各项都为负数．
39It is given that $\mathrm{f}(r) = 1 - \frac{3}{(r+2)!}$ , where $r$ is a non-negative integer. Show that $\mathrm{f}(r) - \mathrm{f}(r-1) = \frac{a(r+1)}{(r+2)!}$ …
40已知 $f(x)= \dfrac {x}{e^{x}}$ ， $f_{1}(x)=f′(x)$ ， $f_{2}(x)=[f_{1}(x)]′$ ， $…$ ， $f_{n+1}(x)=[f_{n}(x)]′$ ， $n∈N^{*}$ ，经…
41The first $n$ terms of a series are given by $\log_a 3 + \log_a 27 + \log_a 243 + \ldots + \log_a 3^{2^{n-1}}$ , where $a$ …
42Ryan has playing cards which he stacks into pyramids. He will begin by stacking up cards to form a pyramid with 1 level,…
43A geometric series has common ratio $r$ , and an arithmetic series has first term $a$ and common difference $d$ , where $a$ …
44At 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…
45A 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…
46已知数列 $\{a_{n}\}$ 的前 $n$ 项和为 $S_{n}$ ， $S_{n}=2a_{n}-3n(n∈N^{*}).(1)$ 证明数列 $\{a_{n}+3\}$ 是等比数列，求出数列 $\{a_{n}\}$ 的通项公式； (2…
47A sequence $u_1, u_2, u_3, \dots$ is such that $u_1 = \frac{1}{3}$ and $u_{r+1} = u_r - \frac{1}{(2r-1)(2r+3)}, \quad \text{for all } r \ge 1.$ …
48A sequence $u_0, u_1, u_2, \dots$ is such that $u_0 = 1$ and $u_n = u_{n-1} + 2n + \frac{1}{n(n+1)}, \quad\text{for all } n \in \mathbb{N}^+.$ …
49The diagrams below show a sequence of patterns formed by squares. Stage 0 is represented by a square of length 1. At eac…
50It 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…
51Prove by induction that, for all positive integers $n$ and for $x \in \mathbb{R}$ , $x \ne 0, x \ne 1$ , $\dfrac{1}{x-1}-\dfrac{1}{x}-\dfrac{1}{x^2}-\dots-\dfrac{1}{x^n}=\dfrac{1}{x^n(x-1)}.$ …
52A rice retailer pledges to donate a bowl of rice for every kilometre run by participants in a service-learning project.…
53The first $n$ terms of a series are given by $\log_a 3 + \log_a 27 + \log_a 243 + \dots + \log_a 3^{2n-1}$ , where $a$ is…
54A sequence $u_1, u_2, u_3, \dots$ is such that $u_1 = -1$ and $u_{n+1} = u_n + 2(3^{-n}) + a$ , where $n \in \mathbb{Z}^+$ …
55A series is given by $\sum_{r=1}^{n} 2(4-3x)^{r}$ where $x$ is constant. Explain why this is a geometric series. Deter…
56The ninth, fifth and second terms of an arithmetic progression are successive terms of a geometric progression with firs…
57The sum of the first $n$ terms of a sequence, $u_r$ , is given by $\displaystyle\sum_{r=1}^n u_r = 1 - \dfrac{n}{(n+1)!}$ …
58Given that $\sum_{r=1}^{n} r^{2} = \frac{n(n+1)(2n+1)}{6},$ find $\sum_{r=1}^{n} (3r+1)^{2}$ in terms of n. Use your…
59A 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…
60已知数列 $\{a_{n}\}$ ， $a_{n}= \dfrac {1}{n(n+2)}(n∈N_{+})$ ，那么 $\dfrac {1}{120}$ 是这个数列的第 ___ 项.