a)cmr:
\(\dfrac{n^5}{5}=\dfrac{n^3}{3}=\dfrac{7n}{15}\) là số nguyên với mọi n \(\in Z\)
b)cmr:với n chẵn thì \(\dfrac{n}{12}+\dfrac{n^2}{8}+\dfrac{n^3}{24}\) là số nguyên
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12 tháng 9 2018
a, Ta có: \(\frac{n^5}{5}+\frac{n^3}{3}+\frac{7n}{15}=\frac{n^5-n}{5}+\frac{n}{5}+\frac{n^3-n}{3}+\frac{n}{3}+\frac{7n}{15}\)
\(=\frac{n^5-n}{5}+\frac{n^3-n}{3}+n\)
Chứng minh \(n^5-n⋮5\Rightarrow\frac{n^5-n}{5}\in Z\)
\(n^3-n⋮3\Rightarrow\frac{n^3-n}{3}\in Z\)
\(\Rightarrow\frac{n^5-n}{5}+\frac{n^3-n}{3}+n\in Z\)
=> Đpcm
b, Tương tự dùng tính chất chia hết
28 tháng 12 2018
\(B=\frac{n^4}{24}+\frac{n^3}{4}+\frac{11n^2}{24}+\frac{n}{4}\)
\(B=\frac{n^4+6n^3+11n^2+6n}{24}\)
\(B=\frac{n^4+2n^3+4n^3+8n^2+3n^2+6n}{24}\)
\(B=\frac{n^3\left(n+2\right)+4n^2\left(n+2\right)+3n\left(n+2\right)}{24}\)
\(B=\frac{\left(n^3+n^2+3n^2+3n\right)\left(n+2\right)}{24}\)
\(B=\frac{n\left(n+1\right)\left(n+3\right)\left(n+2\right)}{24}\)
Lập luận là ra
ST
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GH
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11 tháng 12 2021
\(S=\left(1-\dfrac{1}{4}\right)+\left(1-\dfrac{1}{9}\right)+\left(1-\dfrac{1}{16}\right)+...+\left(1-\dfrac{1}{n^2}\right)\\ S=\left(1+1+...+1\right)-\left(\dfrac{1}{4}+\dfrac{1}{9}+...+\dfrac{1}{n^2}\right)\\ S=n-1-\left(\dfrac{1}{4}+\dfrac{1}{9}+...+\dfrac{1}{n^2}\right)< n-1\)
Lại có \(\dfrac{1}{4}+\dfrac{1}{9}+..+\dfrac{1}{n^2}=\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{n^2}\)
\(\Rightarrow\dfrac{1}{4}+\dfrac{1}{9}+...+\dfrac{1}{n^2}< \dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{n\left(n-1\right)}< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{n-1}-\dfrac{1}{n}=1-\dfrac{1}{n}< 1\)
\(\Rightarrow S>n-1-1=n-2\\ \Rightarrow n-2< S< n-1\\ \Rightarrow S\notin N\)
\(\frac{a^5}{5}+\frac{a^3}{3}+\frac{7a}{15}\left(n\Rightarrow a\text{ }nha\right)=\frac{a^5}{5}+\frac{a^3}{3}+\frac{7a}{15}=\frac{a^5}{5}+\frac{a^3}{3}+\frac{15a-5a-3a}{15}=\frac{a^5-a}{5}+\frac{a^3-a}{3}+\frac{15a}{15}=\frac{a^5-a}{5}+\frac{a^3-a}{3}+a;a^k-a⋮k\left(a\in Z;1< k\in N\right)\left(fecmat\right)\Rightarrow\left\{{}\begin{matrix}a^5-a⋮5\\a^3-a⋮3\end{matrix}\right.\Rightarrow dpcm\)
\(\frac{a}{12}+\frac{a^2}{8}+\frac{a^3}{24}\left(n\Rightarrow a\text{ nha}\right)=\frac{a^3+3a^2+2a}{24}=\frac{\left(a+2\right)\left(a+1\right)a}{24}.a=2k\left(k\in N\right)\Rightarrow;\frac{a\left(a+1\right)\left(a+2\right)}{24}=\frac{2k.\left(2k+1\right)\left(2k+2\right)}{24}=\frac{k\left(k+1\right)\left(2k+1\right)}{6}\Leftrightarrow k\left(k+1\right)\left(2k+1\right)⋮6\)