Math topic中文

Geometric series

Formula for the $n$ th term and the sum of a finite geometric 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…
3等比数列 $\{a_{n}\}$ 中，若 $a_{2}$ 、 $a_{4}$ 是方程 $2x^{2}-11x+8=0$ 的两根，则 $a_{3}$ 的值为 $(\quad)$ A. $2$ B. $±2$ C. $\sqrt{2}$ …
4The sum of the first 100 terms of an arithmetic progression is 10,000. The first, second and fifth terms of this progres…
5A 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…
6Prove by induction that, for all positive integers $n$ and for $x\in\mathbb{R}$ , $x\ne 0,\ x\ne 1$ , $\frac{1}{x-1}-\frac{1}{x}-\frac{1}{x^2}-\dots-\frac{1}{x^n}=\frac{1}{x^n(x-1)}\,.$ …
7等差数列 $\{a_{n}\}$ 的前 $n$ 项和为 $S_{n}$ ，数列 $\{b_{n}\}$ 为等比数列， $S_{n}\text{=}n^{2}\text{+}n$ ， $b_{1}\text{=}a_{1}\text{,}b_{2}\text{=}\dfrac{a_{1}a_{2}}{2}\text{.}$ …
8A 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…
9已知数列 $\{a_{n}\}$ 是一个首项为 $a_{1}$ ，公比 $q > 0$ 的等比数列，前 $n$ 项和为 $S_{n}$ ，记 $T_{n}=a_{1}+a_{2}+a_{3}+…+a_{2n-1}$ ，求 $\lim\limits_{n→∞} \dfrac {S_{n}}{T_{n}}$ …
10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 $a_{n}^{2}={{2}^{{{b}_{n}}}}$ …
11An arithmetic sequence has first term $a$ and common difference $0.5$ , where $a$ is an integer. Find the smallest value…
12Anand writes a computer programme to simulate a population of organisms in a controlled environment. It is assumed that…
13On the first day of last month, the temperature of a machine in a manufacturing plant was found to be $90^\circ\text{C}$ …
14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. $- 32$ …
15An arithmetic progression has first term $a$ and common difference $d$ , where $a$ and $d$ are non-zero. The first, third…
16Aenicillin, a type of bacteria, is grown in a petri dish. Its population doubles every six hours. Denoting the amount of…
17已知首项为 $1$ 的数列 $\{a_{n}\}$ 的前 $n$ 项和为 $S_{n}$ ，且 $a_{n\text{+}1}\text{=}\dfrac{1}{3}S_{n}$ ，数列 $\{b_{n}\}$ 满足 $b_{n}\text{=}\log_{3}a_{n}$ …
18记 $S_{n}$ 为递增等比数列 $\{a_{n}\}$ 的前 $n$ 项和，若 $a_{1}a_{2}a_{3}=8$ ， $a_{4}=a_{3}+4$ ，则 $S_{10}$ 的值为 ___ .
19已知三个实数 $2$ ， $m$ ， $8$ 构成一个等比数列，则圆锥曲线 $\dfrac {x^{2}}{m}+ \dfrac {y^{2}}{2}=1$ 的离心率为 $()$ A. $\dfrac { \sqrt {2}}{2}$ …
20A 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…
21It is given that $f(r)=(r-2)2^{r-1}$ . Show that $f(r+1)-f(r)=2^r a r$ , where $a$ is a constant to be determined. Hence f…
22A team of cyclists plans to ride from Singapore to Penang and back over 10 days. The distance for the route taken from S…
23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 $\{ \dfrac {1}{s_{n}}\}$ …
24Betty received an SMS from a friend regarding a Bitcoin (BTC) investment that guarantees a fixed percentage of interest…
25Adam and Gregory signed up for a marathon. In preparation for this marathon, Adam and Gregory each planned a 15-week per…
26Jane decides to purchase an apartment. To finance her apartment, she intends to apply for a bank housing loan of $847 5…
27已知 $S_{n}$ 为数列 $\{a_{n}\}$ 的前 $n$ 项和， $S_{n}=1-a_{n}$ ，则 $S_{5}=(\quad)$ A. $\dfrac{31}{16}$ B. $\dfrac{31}{2}$ C.…
28By writing $\dfrac{2}{r(r^2-1)}$ in partial fractions, find an expression for $\displaystyle\sum_{r=2}^{n}\dfrac{2}{r(r^2-1)}$ …
29Dendrologists are specialised scientists who study trees and woody plants. Their work is diverse and can encompass vario…
30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…
31In a computer simulation, each player uses an identical pail to pour water repeatedly into his assigned tank, which is i…
32设等比数列 $\{a_{n}\}$ 的前 $n$ 项和为 $S_{n}.$ 若 $S_{2}=3$ ， $S_{4}=15$ ，则 $S_{6}=(\quad)$ A. $31\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$ …
33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…
34Wendie attends training sessions in preparation for a running race. In the first session, she runs a distance of 1000 me…
35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 $\{b_{n}\}$ …
36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 $b_{n}=a_{n+1}-a_{n}$ …
37A 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…
38Selena decides to borrow $200 000 from the bank to fund her home purchase on 1 January 2025. On the first day of each m…
39The reproduction number, $R_0$ , of a disease is a mathematical term that indicates how contagious an infectious disease…
40A particular set of Russian dolls consists of seven dolls. The heights of the seven dolls are in arithmetic progression…
41Cadence, the pedal speed in cycling, is measured in pedal stroke revolutions per minute (RPM). For example, a cadence of…
42In 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…
43A small trading company made a profit of $250000 in the year 2000. The company considered two different plans, plan A a…
44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 $n\geqslant 2$ ， $n∈N^{*}$ …
45In 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…
46己知数列 $\{a_{n}\}$ 为等比数列，且 $a_{2}a_{3}a_{4}=-a_{7}^{2}=-64$ ，则 $\tan (\dfrac{{{a}_{4}}{{a}_{6}}}{3}\cdot \mathrm{ }\!\!\pi\!\!{ })=()$ …
47A financial institution, Future Investments Inc., has introduced a new investment scheme. The scheme pays a compound int…
48在等比数列 $\{a_{n}\}$ 中， $a_{n}\in R$ ， $a_{3}$ ， $a_{11}$ 是方程 $3x^{2}-25x+27=0$ 的两根，则 $a_{7}=$ ___.
49A hospital patient is given a pill containing 50 units of antibiotic at 12pm. By 1pm, the number of units of antibiotic…
50An arithmetic series has first term $a$ and common difference $d$ , where $a$ and $d$ are non-zero. If the first, fifth a…
51已知数列 $\{a_{n}\}$ 为等比数列， $a_{1}=1$ ， $a_{4}=8$ ，则 $\{a_{n}\}$ 的前 $5$ 项和 $S_{5}=$ ___ ．
52A data processing centre processes data (in suitable units) every day. Let $u_n$ , where $n$ is an integer, $n \ge 1$ , re…
53设数列 $\{a_{n}\}$ 满足 $a_{n+1}= \dfrac {4a_{n}-2}{a_{n}+1}(n∈N^{*})(1)$ 若 $a_{1}=3$ ， $b_{n}= \dfrac {2-a_{n}}{a_{n}-1}(n∈N^{*})$ …
54A sequence $u_1,u_2,u_3,\dots$ is such that $u_1 = \dfrac{3}{4}$ and $u_{n+1} = u_n + \dfrac{3}{4}\left(\dfrac{1}{2}\right)^{2n}$ …
55It is given that $a$ , $b$ , $c$ are the first three terms of a geometric progression. It is also given that $a$ , $c$ , $b$ …
56An arithmetic series has positive first term $a$ and common difference $d$ . The first, fifth and second terms of the ari…
57A hospital patient is given a pill containing 50 units of antibiotic at 12pm. By 1pm, the number of units of antibiotic…
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 $a_{1}\text{=}1$ ， $a_{2}\text{=}2$ ， ${a}_{n+2}=(1+{{\rm\cos}}^{2}\dfrac{nπ}{2}){a}_{n}+{{\rm\sin}}^{2}\dfrac{nπ}{2}$ …