rút gọn các phân thức vs n là số tự nhiên:
a,\(\frac{\left(n+1\right)!}{n!\left(n+2\right)}\)b,\(\frac{n!}{\left(n+1\right)!-n!}\)c,\(\frac{\left(n+1\right)!-\left(n+2\right)!}{\left(n+1\right)+\left(n+2\right)!}\)
a) \(\frac{\left(n+1\right)!}{n!\left(n+2\right)}=\frac{n!\left(n+1\right)}{n!\left(n+2\right)}=\frac{n+1}{n+2}\)
b)\(\frac{n!}{\left(n+1\right)!-n!}=\frac{n!}{n!\left(n+1\right)-n!}=\frac{n!}{n!\left(n+1-1\right)}=\frac{1}{n}\)
c)\(\frac{\left(n+1\right)!-\left(n+2\right)!}{\left(n+1\right)!+\left(n+2\right)!}=\frac{n!\left(n+1\right)-n!\left(n+1\right)\left(n+2\right)}{n!\left(n+1\right)+n!\left(n+1\right)\left(n+2\right)}=\frac{n!\left(n+1\right)\left(1-n-2\right)}{n!\left(n+1\right)\left(1+n+2\right)}=\frac{-n-1}{n+3}\)
( Kí hiệu n!=1.2.3.4...n)
Rút gọn biểu thức sau với n là số tự nhiên:
\(\left(1+\frac{2}{4}\right).\left(1+\frac{2}{10}\right).\left(1+\frac{2}{18}\right)...\left(1+\frac{2}{n^2+3n}\right)\)
Đặt A = \(\left(1+\frac{2}{4}\right).\left(1+\frac{2}{10}\right).\left(1+\frac{2}{18}\right).....\left(1+\frac{2}{n^2+3n}\right)\)
Ta có : A = \(\left(1+\frac{2}{4}\right).\left(1+\frac{2}{10}\right).\left(1+\frac{2}{18}\right).....\left(1+\frac{2}{n^2+3n}\right)\)
= \(\frac{6}{4}.\frac{12}{10}.\frac{20}{18}.....\frac{\left(n+1\right).\left(n+2\right)}{n.\left(n+3\right)}\)
= \(\frac{3.2}{4}.\frac{3.4}{2.5}.\frac{4.5}{3.6}.....\frac{\left(n+1\right).\left(n+2\right)}{n.\left(n+3\right)}\)
= \(\frac{3.2.3.4.4.5....n}{2.3.4.5.6.....\left(n+2\right)}\)
= \(\frac{3.\left(n+1\right)}{n+2}\)
Vậy A = \(\frac{3.\left(n+1\right)}{n+2}\)
1. A= \(\left(\sqrt{x}-\frac{x+2}{\sqrt{x}-1}\right):\left(\frac{\sqrt{x}}{\sqrt{x}+1}-\frac{\sqrt{x}-4}{1-x}\right)\)
a. Rút gọn A
b. Tìm x để A<0
c. Tìm giá trị nhỏ nhất A.
2. M=\(\left(\frac{2x+1}{\sqrt{x^3}-1}-\frac{1}{\sqrt{x}-1}\right):\left(1+\frac{x+4}{x+\sqrt{x}+1}\right)\)
a. Rút gọn M
b. Tìm số nguyên x để M có giá trị nguyên
3. N=\(\left(\frac{\sqrt{a}+\sqrt{b}}{1-\sqrt{a.b}}+\frac{\sqrt{a}-\sqrt{b}}{1+\sqrt{a.b}}\right):\left(1+\frac{a+b+2ab}{1-ab}\right)\)
a. Rút gọn N
b. Tính N khi a=\(\frac{2}{2-\sqrt{3}}\)
c. Tìm số nguyên a để N có giá trị nguyên
Gíup mình với. Cảm ơn nhiều ạ.
Rút gọn rồi tìm các giá trị nguyên của x để A là số nguyên
\(A=\left(1-\frac{1}{x+1}\right).\left(1-\frac{1}{x+2}\right).\left(1-\frac{1}{x+3}\right)...\left(1-\frac{1}{x+2021}\right).\left(1-\frac{1}{x+2022}\right)\)
1. Tìm giá trị của x để các phân thức sau bằng 0
a. \(\frac{x^4+x^3+x+1}{x^4-x^3+2x^2-x+1}\)
b. \(\frac{x^4-5x^2+4}{x^4-10x^2+9}\)
2. Rút gọn các phân thức:
a. \(\frac{3x^3-7x^2+5x-1}{2x^3-x^2-4x+3}\) b. \(\frac{\left(x-y\right)^3-3xy\left(x+y\right)+y^{^3}}{x-6y}\)
3. Rút gọn các phân thức với n là số tự nhiên
a. \(\frac{\left(n+1\right)!}{n!\left(n+2\right)}\) b. \(\frac{n!}{\left(n+1\right)!-n!}\) c. \(\frac{\left(n+1\right)!-\left(n+2\right)!}{\left(n+1\right)!+\left(n+2\right)!}\)
Rút gọn phân số
a,\(S=1+\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^n}.\)
b,\(C=\left(\frac{1}{2}+1\right)\left(\frac{1}{3}+1\right)\left(\frac{1}{4}+1\right).....\left(\frac{1}{99}+1\right)\)
c,\(D=\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}...\frac{899}{30^2}\)
1. Tìm các số tự nhiên x, y sao cho: \(7^x+12^y=50\).
2.Tìm tất cả các số tự nhiên n để phân số\(\frac{18n+3}{21n+7}\)có thể rút gọn được
\(\left(1-\frac{1}{2}\right).\left(1-\frac{1}{3}\right)..................\left(1-\frac{1}{20}\right)\)
=\(\frac{1}{2}.\frac{2}{3}.............\frac{19}{20}\)
=\(\frac{1.2.3..............19}{2.3.4..............20}\)
=\(\frac{1}{20}\)
Rút gọn biểu thức
\(A=\left(1-\frac{1}{2^2}\right).\left(1-\frac{1}{3^2}\right).\left(1-\frac{1}{4^2}\right)...\left(1-\frac{1}{n^2}\right)\)
\(A=\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right)...\left(1-\frac{1}{n^2}\right)\)
\(=\left(\frac{2^2-1}{2^2}\right)\left(\frac{3^2-1}{3^2}\right)\left(\frac{4^2-1}{4^2}\right)...\left(\frac{n^2-1}{n^2}\right)\)
\(=\text{[}\frac{\left(2-1\right)\left(2+1\right)}{2^2}\text{]}.\text{[}\frac{\left(3-1\right)\left(3+1\right)}{3^2}\text{]}.\text{[}\frac{\left(4-1\right)\left(4+1\right)}{4^2}\text{]}...\text{[}\frac{\left(n-1\right)\left(n+1\right)}{n^2}\text{]}\)
\(=\left(\frac{1.3}{2^2}\right).\left(\frac{2.4}{3^2}\right).\left(\frac{3.5}{4^2}\right)...\text{[}\frac{\left(n-1\right)\left(n+1\right)}{n^2}\text{]}\)
\(=\frac{\text{[}1.2.3...\left(n-1\right)\text{]}.\text{[}3.4.5...\left(n+1\right)\text{]}}{\text{[}2.3.4...n\text{]}.\text{[}2.3.4...n\text{]}}\)
\(=\frac{1}{n}.\frac{n+1}{2}\)
\(=\frac{n+1}{2n}\)
