Math topic中文

Binomial theorem

use of the Binomial Theorem for positive integer $n$

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…
3Find, in ascending powers of $x$ , the first three terms in the expansion of $(1+4x)^7$ , $(2-x)^7$ . Hence, find the coeff…
4The first two non-zero terms in the expansion of $(1 + bx)(1 + ax)^6$ are $1$ and $-\frac{21}{4}x^2$ . Given that $a < b$ …
5二项式 $(3 \sqrt[3]{x}+ \dfrac {1}{x})^{n}$ 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．
6Explain why the binomial expansion of $\left(ax + \dfrac{2}{x}\right)^6$ contains only even powers of $x$ . Write down, a…
7 $(x+y)^{2}(x-2y)^{4}$ 的展开式中 $x^{2}y^{4}$ 的系数为 $(\quad)$ A. $88$ B. $104$ C. $-40$ D. $-24$
8Write down the first three terms in the expansion, in descending powers of x, of $\left(x^{2}+\dfrac{m}{x}\right)^{9}$ ,…
9Write down the first four terms in the expansion of $\left(1 - 4x\right)^{6}$ . Hence find the coefficient of $x^{3}$ in…
10 $(2x-\dfrac{1}{x})^{5}(x+\dfrac{1}{x})$ 的展开式中的常数项为 $(\quad)$ A. $20$ B. $-20$ C. $40$ D. $-40$
11Write down and simplify the first three terms in the expansion, in descending powers of x, of $\left(2 - \dfrac{3}{x}\right)^8$ …
12State 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…
13The first two non-zero terms in the expansion of $(1 + bx)(1 + ax)^6$ in ascending powers of $x$ are $1$ and $-\frac{21}{4}x^2$ …
14Write down, and simplify, the first 3 terms in the expansion of $\left(2 - x\right)^7$ in ascending powers of $x$ . Find…
15若 $(4x-7)^{6}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots+a_{6}x^{6}$ ，则 $a_{1}+a_{3}+a_{5}=(\quad)$ A. $\dfrac{7^{6}-11^{6}}{2}$ …
16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 ${a}_{1}+{a}_{2}+…{a}_{7}$ …
17Find the first 4 terms in the expansion of $\left(\frac{a}{x}+3x^2\right)^6$ in ascending power of $x$ , simplifying each…
18关于 $(2x+\dfrac{1}{x})^{12}$ 的展开式，下列结论正确的是 $(\quad)$ A. 所有二项式系数和为 $2^{12}$ B. 所有项的系数和为 $3^{12}$ C. 二项式系数最大的项为第 $6$ 项 D…
19已知 $a$ 是 $A=7^{n}+C_{n}^{1}7^{n-1}+C_{n}^{2}7^{n-2}+⋯+C_{n}^{n-1}7(n$ 为正奇数 $)$ 被 $4$ 整除的余数，则 $∫_{1}^{e}(x+\dfrac{a}{x}){\rm d}x$ …
20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 $x^{2}$ …
21Find the first 4 terms in the expansion of $\left(2 - \dfrac{3}{x^{2}}\right)^{7}$ in descending powers of $x$ , simplify…
22已知函数 $f(x)=-\dfrac{1}{8}x+3\cos 2x$ 在 $x=0$ 处的切线与直线 $y=nx$ 垂直，若 $(1+x+x^{2})(1+x)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+⋯+a_{n+2}x^{n+2}$ …
23已知 $(x-m)^{7}=a_{0}+a_{1}x+a_{2}x^{2}+…+a_{7}x^{7}$ 的展开式中 $x^{4}$ 的系数是 $-35$ ，则 $a_{1}+a_{2}+a_{3}+…a_{7}=$ ___ ．
24State 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…
25(1)求证： $1+2+2^{2}+$ … $+2^{5n-1}(n\in N^{*})$ 能被 $31$ 整除； (2)求 $S=C_{27}^{1}+C_{27}^{2}+⋯+C_{27}^{27}$ 除以 $9$ 的余数．
26Write down and simplify the first three terms in the expansion, in ascending powers of $x$ , of $\left(2 + \frac{x}{2}\right)^n$ …
27Write down the first 3 terms in the expansion of $\left(2 - \frac{x}{3}\right)^6$ , in ascending power of $x$ . All terms…
28Expand and simplify $\left(\frac{1}{2}-2x\right)^5$ in ascending powers of $x$ , up to the first 4 terms. Hence find the…
29在 $(x-\dfrac{2}{x^{2}})^{5}$ 的展开式中， $x^{-1}$ 的系数是 ___ $.($ 用数字作答 $)$
30已知 $(x+\dfrac{4}{x}-4)^{6}=a_{1}x^{6}+a_{2}x^{5}+⋯+a_{7}+⋯+a_{13}x^{-6}$ ，则 $a_{2}=$ ___， $6a_{1}+5a_{2}+⋯+a_{6}-a_{8}-2a_{9}-⋯-6a_{13}=$ …
31已知数列 $\{a_{n}\}$ 满足 $a_{1}=\dfrac{1}{k}$ ， $k\geqslant 2$ ， $k\in N^{*}$ ， $[a_{n}]$ 表示不超过 $a_{n}$ 的最大整数 $($ 如 $[1.6]=1)$ …
32 $(1+2x)^{5}$ 的展开式中，各项二项式系数的和是 $(\quad)$ A. $1$ B. $-1$ C. $2^{5}$ D. $3^{5}$
33It is given that the first three terms, in ascending powers of $x$ , of the binomial expansion of $(2+ax)^6$ are $64 - 960x + bx^2$ …