Math topic中文

Arithmetic series

Formula for the $n$ th term and the sum of a finite arithmetic series.

Practice questions

1The sum of the first $n$ terms of a series is given by $S_n = 3n(n + 2)$ . Show that this series follows an arithmetic pr…
2The sum of the first $n$ terms of a series is given by $S_n = 3n(n + 2)$ . Show that this series follows an arithmetic pr…
3The sum of the first 100 terms of an arithmetic progression is 10,000. The first, second and fifth terms of this progres…
4已知 $\left\{{a}_{n}\right\}$ 是公差为 $2$ 的等差数列， ${S}_{n}$ 为 $\left\{{a}_{n}\right\}$ 的前 $n$ 项和，若 $S_{5}=S_{3}$ ，则 $a_{4}=(\quad)$ …
5等差数列 $\{a_{n}\}$ 和 $\{b_{n}\}$ 的前 $n$ 项和分别为 $S_{n}$ 和 $T_{n}$ ，且 $\dfrac{S_{n}}{T_{n}}=\dfrac{3n}{2n+3}$ ，则 $\dfrac{a_{5}}{b_{5}}=(\quad)$ …
6A 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…
7有一堆电线杆，从它的截面看是一个近似的三角形，最底层有33根，相邻两层相差2根．这堆电线杆共有多少根？
8已知定义：在数列 $\{a_{n}\}$ 中，若 $a_{n}^{2}-a_{n-1}^{2}=p(n\geqslant 2,n∈N^{*},p$ 为常数 $)$ ，则称数列 $\{a_{n}\}$ 为等方差数列，下列判断： $①\{(-1)^{n}\}$ …
9等差数列 $\{a_{n}\}$ 中， $a_{n} > 0$ ， $a_{1}^{2}+a_{7}^{2}+2a_{1}a_{7}=4$ ，则它的前 $7$ 项的和等于 $()$ A. $\dfrac {5}{2}$ B. $5$ …
10The sum, $S_n$ , of the first $n$ terms of the sequence $u_1, u_2, u_3, \dots$ is given by $S_n = n^2 + 4n .$ Find $u_n$ …
11数列 ${-}1{,}5{,}{-}9{,}13{,}{…}$ 的一个通项公式是 $a_{n}{=}$ ___ ．
12已知数列 $\{a_{n}\}$ 的前 $n$ 项和为 $S_{n}$ ，首项为 $a_{1}$ ，且 $4$ ， $a_{n}$ ， $S_{n}$ 成等差数列． (1)求数列 $\{a_{n}\}$ 的通项公式； (2)若 $a_{n}^{2}={{2}^{{{b}_{n}}}}$ …
13An arithmetic sequence has first term $a$ and common difference $0.5$ , where $a$ is an integer. Find the smallest value…
14Anand writes a computer programme to simulate a population of organisms in a controlled environment. It is assumed that…
15已知等比数列 $\{a_{n}\}(a_{1}\neq a_{2})$ 的公比为 $q$ ，且 $a_{7}$ ， $a_{1}$ ， $a_{4}$ 成等差数列，则 $q=()$ A. $1$ 或 $- 32$ B. $- 32$ …
16An arithmetic progression has first term $a$ and common difference $d$ , where $a$ and $d$ are non-zero. The first, third…
17Aenicillin, a type of bacteria, is grown in a petri dish. Its population doubles every six hours. Denoting the amount of…
18 $\sqrt {2}+1$ 与 $\sqrt {2}-1$ 的等差中项是 $()$ A. $1$ B. $-1$ C. $\sqrt {2}$ D. $±1$
19A finite sequence $\{a_n\}$ has 50 terms and is such that $a_{n+1} = a_n + 0.15$ for $n = 1,2,3,\ldots,49$ . Another infi…
20若 $2x$ ， $x+1$ ， $x+2$ 成等差数列， $x=$ ___ ．
21A team of cyclists plans to ride from Singapore to Penang and back over 10 days. The distance for the route taken from S…
22已知在数列 $\{a_{n}\}$ 中， $a_{1}=1$ ，其前 $n$ 项和为 $s_{n}$ ，且 $a_{n}= \dfrac {2 s_{ n }^{ 2 }}{2s_{n}-1}(n\geqslant 2)(1)$ 证明 $\{ \dfrac {1}{s_{n}}\}$ …
23Adam and Gregory signed up for a marathon. In preparation for this marathon, Adam and Gregory each planned a 15-week per…
24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， $n$ …
25Dendrologists are specialised scientists who study trees and woody plants. Their work is diverse and can encompass vario…
26已知等差数列 $\{a_{n}\}$ 的前 $n$ 项和为 $S_{n}(n∈N^{*})$ ，且 $a_{n}=2n+λ$ ，若数列 $\{S_{n}\}$ 在 $n\geqslant 7$ 时为递增数列，则实数 $λ$ 的取值范围为 $()$ …
27市民小张计划贷款 $60$ 万元用于购买一套商品住房，银行给小张提供了两种贷款方式.①等额本金：每月的还款额呈递减趋势，且从第二个还款月开始，每月还款额与上月还款额的差均相同；②等额本息：每个月的还款额均相同.银行规定，在贷款到账日的次月当…
28Find how many positive integers, less than 1000, are odd numbers. odd numbers which are not divisible by 5. Find the sum…
29已知公差不为 $0$ 的等差数列 $\{a_{n}\}$ 的前 $n$ 项和为 $S_{n}$ ，且 $2S_{n}=a_{n+1}a_{n}$ ，则 $a_{1}=$ ___ .
30In a computer simulation, each player uses an identical pail to pour water repeatedly into his assigned tank, which is i…
31Find the exact value of $\pi \int_{1}^{6} x^3 \text{ d}x$ . The diagram shows a sketch of the curve $y = x^{\frac{3}{2}}$ …
32已知等差数列 $\{a_{n}\}$ 的前 $n$ 项和为 $S_{n}$ ，且 $a_{1}=d=9$ ，若 $S_{3n}-S_{2n}=S_{2n}-S_{n}+729$ ，则 $n$ 的值为 $(\quad)$ A. $8$ …
33A circular cake of radius $r$ is cut into 22 sectors. The areas of the sectors form an increasing arithmetic progression…
34Find $\int \frac{x}{\sqrt{2x-1}}\,dx.$ Using the substitution $t = \tan x$ , find $\int \frac{1}{4\cos^{2}x + 9\sin^{2}x}\,dx.$ …
35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___.
36Wendie attends training sessions in preparation for a running race. In the first session, she runs a distance of 1000 me…
37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 $\{b_{n}\}$ …
38设 $S_{n}$ 是等差数列 $\{a_{n}\}$ 的前 $n$ 项和，若 $\dfrac{S_{3}}{S_{6}}=\dfrac{1}{3}$ ，则 $\dfrac{S_{9}}{S_{6}}=$ ___ .
39已知各项均为正数的数列 $\{a_{n}\}$ 的前 $n$ 项和为 $S_{n}$ ，且 $a_{1}=1$ ， $a_{n+1}(a_{n+1}-1)=a_{n}(a_{n}+1).$ 若 $[x]$ 表示不超过 $x$ 的最大整数， $b_{n}=[\dfrac{(n+1)^{2}}{2S_{n}}]$ …
40A finite sequence $\{a_n\}$ has 50 terms and is such that $a_{n+1} = a_n + 0.15$ for $n = 1,2,3,\ldots,49$ . Another infi…
41The reproduction number, $R_0$ , of a disease is a mathematical term that indicates how contagious an infectious disease…
42A particular set of Russian dolls consists of seven dolls. The heights of the seven dolls are in arithmetic progression…
43Cadence, the pedal speed in cycling, is measured in pedal stroke revolutions per minute (RPM). For example, a cadence of…
44In a chemical reaction, a compound X is formed from two compounds Y and Z. The masses in grams of X, Y and Z present at…
45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. $2n\:\:\:\:\:\:\:\:\:\:$ …
46A small trading company made a profit of $250000 in the year 2000. The company considered two different plans, plan A a…
47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 $n\geqslant 2$ ， $n∈N^{*}$ …
48已知等差数列 $\{a_{n}\}$ 的前 $n$ 项和为 $S_{n}$ ，若 $a_{2}=7$ ， $a_{6}+a_{8}=-6$ ，则 $S_{n}$ 取最大值时， $n$ 的值为 $()$ A. $3$ B. $4$ …
49In an art lesson, wires are cut and bent to create a series of shapes. Student A cuts a piece of wire of length 100 cm t…
50Find $\sum_{r=0}^{n} [(n+2)r + n]$ , giving your answer in terms of $n$ . [You may use the result $\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2$ …
51设等差数列 $\{a_{n}\}$ 的前 $n$ 项和为 $S_{n}$ ，若 $S_{7}=21$ ， $S_{21}=33$ ，则 $S_{14}=(\quad)$ A. $27$ B. $45$ C. $32$ D. $11$ …
52A financial institution, Future Investments Inc., has introduced a new investment scheme. The scheme pays a compound int…
53An arithmetic series has first term $a$ and common difference $d$ , where $a$ and $d$ are non-zero. If the first, fifth a…
54等差数列 $\{a_{n}\}$ 的前 $n$ 项和为 $S_{n}$ ，若 $S{\,\!}_{2}=2$ ， $S{\,\!}_{4}=10$ ，则 $S{\,\!}_{6}$ 等于 $()$ A. $12$ B.…
55A data processing centre processes data (in suitable units) every day. Let $u_n$ , where $n$ is an integer, $n \ge 1$ , re…
56It is given that $a$ , $b$ , $c$ are the first three terms of a geometric progression. It is also given that $a$ , $c$ , $b$ …
57An arithmetic series has positive first term $a$ and common difference $d$ . The first, fifth and second terms of the ari…
58Sarah, a Singaporean Citizen, started a 4-year undergraduate Computing course at the National University of Singapore (N…
59An arithmetic series has positive first term $a$ and common difference $d$ . The first, fifth and second terms of the ari…
60设数列 $\{a_{n}\}$ 满足 $a_{1}\text{=}1$ ， $a_{2}\text{=}2$ ， ${a}_{n+2}=(1+{{\rm\cos}}^{2}\dfrac{nπ}{2}){a}_{n}+{{\rm\sin}}^{2}\dfrac{nπ}{2}$ …