Math topic中文

General term

use of the general term $\binom{n}{r} a^{n-r} b^{r}$ , $0 \leqslant r \leqslant n$ (knowledge of the greatest term and properties of the coefficients is not required)

Practice questions

1Explain why all terms in the expansion of $\left(kx^3 + x\right)^{12}$ do not contain any odd powers of $x$ . Given that…
2It is given that $\left(x + \dfrac{k}{x^2}\right)^n$ is a binomial expansion where $k$ and $n$ are positive constants. W…
3Calculate the term independent of $x$ in the binomial expansion of $\left(x - \frac{1}{2x^5}\right)^{18}$ .
4二项式 $(3 \sqrt[3]{x}+ \dfrac {1}{x})^{n}$ 的展开式中的各项系数和为 $P$ ，所有二项式系数和为 $Q$ ，若 $P+Q=272$ ，求展开式中的常数项．
5Explain why the binomial expansion of $\left(ax + \dfrac{2}{x}\right)^6$ contains only even powers of $x$ . Write down, a…
6By considering the general term in the binomial expansion of $\left(x + \frac{1}{ax^2}\right)^7$ , where $a$ is a constan…
7In the expansion of $\left(x^2 - \frac{m}{2x}\right)^{12}$ , where $m$ is a positive constant, the independent term of $x$ …
8Write down the general term, $T_{r+1}$ , for the binomial expansion of $\left(2x + \dfrac{1}{2x^3}\right)^{12}$ . Find the…
9Write down the general term in the binomial expansion of $\left(x + \frac{5}{x}\right)^n$ . Given that there is a constan…
10Write down the first three terms in the expansion, in descending powers of x, of $\left(x^{2}+\dfrac{m}{x}\right)^{9}$ ,…
11Write down the first four terms in the expansion of $\left(1 - 4x\right)^{6}$ . Hence find the coefficient of $x^{3}$ in…
12 $(2x-\dfrac{1}{x})^{5}(x+\dfrac{1}{x})$ 的展开式中的常数项为 $(\quad)$ A. $20$ B. $-20$ C. $40$ D. $-40$
13By considering the general term in the binomial expansion of $\left(x^{4}-\frac{1}{k x^{2}}\right)^{6}$ , where $k$ is a…
14State the third term in the expansion of $(2+x)^n$ in terms of $a$ and $n$ . Write down the first three terms in the expa…
15By considering the general term in the binomial expansion of $\left(x^2 - \frac{1}{3x}\right)^9$ , explain why the powers…
16Write down, and simplify, the first 3 terms in the expansion of $\left(2 - x\right)^7$ in ascending powers of $x$ . Find…
17By considering the general term in the expansion of $\left(\dfrac{2}{x^3}-x^2\right)^{10}$ , explain why there is no term…
18By considering the general term in the binomial expansion of $\left(x^2 - \frac{1}{2x}\right)^{17}$ , explain why there i…
19 $(2x-\dfrac{1}{\sqrt{x}})^{6}$ 的展开式中的常数项等于___.
20In the binomial expansion of $\left(x + \frac{k}{x}\right)^5$ , where $k$ is a positive integer, the coefficients of $\frac{1}{x^3}$ …
21已知 $\left(2x+1\right){\left(x-2\right)}^{6}={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+…{a}_{7}{x}^{7}$ ． $($ Ⅰ $)$ 求 ${a}_{1}+{a}_{2}+…{a}_{7}$ …
22Write down the general term in the binomial expansion of $\left(px - \dfrac{q}{x}\right)^n$ , where $p$ and $q$ are posit…
23In the expansion of $(2x-1)^2\left(1+\dfrac{p}{x}\right)^8$ , where $p$ is a positive constant, there is no term in $\dfrac{1}{x^3}$ …
24Find the first 4 terms in the expansion of $\left(\frac{a}{x}+3x^2\right)^6$ in ascending power of $x$ , simplifying each…
25设 $a= \int _{ 0 }^{ π }(\sin x+\cos x)dx$ ，且二项式 $(a \sqrt {x}- \dfrac {1}{ \sqrt {x}})^{n}$ 的所有二项式系数之和为 $64$ ，则其展开式中含 $x^{2}$ …
26Find the first 4 terms in the expansion of $\left(2 - \dfrac{3}{x^{2}}\right)^{7}$ in descending powers of $x$ , simplify…
27It is given that there is a term in $x^2$ in the expansion of $\left(\sqrt{x} - \frac{2}{x}\right)^n$ , where $n$ is a po…
28State the third term in the expansion of $(2+x)^n$ in terms of $a$ and $n$ . Write down the first three terms in the expa…
29In the binomial expansion of $\left(x + \frac{k}{x}\right)^5$ , where $k$ is a positive constant, the coefficients of $x^3$ …
30Find the term independent of $x$ and the $\frac{1}{x^3}$ term in the binomial expansion of $\left(x^2 - \frac{2}{x}\right)^9.$ …
31Write down and simplify the first three terms in the expansion, in ascending powers of $x$ , of $\left(2 + \frac{x}{2}\right)^n$ …
32In the expansion of $\left(x^2 - \frac{1}{2x^4}\right)^n$ , the sixth term is independent of $x$ . Show that the value of…
33By considering the general term in the binomial expansion of $\left(\frac{2}{x^2} + x^3\right)^{16}$ , explain why every…
34Write down the first 3 terms in the expansion of $\left(2 - \frac{x}{3}\right)^6$ , in ascending power of $x$ . All terms…
35Write down, and simplify, the general term in the binomial expansion of $\left(x^2 + \frac{3}{x}\right)^{15}$ . Hence det…
36Given that the constant term in the binomial expansion of $\left( x - \dfrac{k}{x^{3}} \right)^{8}$ is 7, find the value…
37It is given that the first three terms, in ascending powers of $x$ , of the binomial expansion of $(2+ax)^6$ are $64 - 960x + bx^2$ …