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\(\text{Ta có: }\)
\(VP=\frac{1}{a.\left(a+1\right)}=\frac{a+1-a}{a.\left(a+1\right)}=\frac{a+1}{a.\left(a+1\right)}-\frac{a}{a.\left(a+1\right)}=\frac{1}{a}-\frac{1}{a+1}=VT\left(đpcm\right)\)
trong phân số mẫu luôn thuộc Z và lớn hơn 0
nên a ∈ Z và a ≠ 0
\(\frac{1}{a+1}\)nếu a=-1 thì \(\frac{1}{-1+1}\)=\(\frac{1}{0}\)mẫu khác 0 nên a ≠ -1
\(\text{Vì }\left[a,b\right],\left[b,c\right],\left[c,a\right]\text{ là BCNN}\)
\(\Rightarrow\left[a,b\right]=a.b;\left[b,c\right]=b.c;\left[c,a\right]=c.a\)
\(\Rightarrow\frac{1}{\left[a+b\right]}+\frac{1}{\left[b+c\right]}+\frac{1}{\left[c+a\right]}=\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\)
\(\text{Giả sử }a< b< c\)
\(\Rightarrow a\le2;b\le3;c\le5\)
\(\Rightarrow\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\le\frac{1}{2.3}+\frac{1}{3.5}+\frac{1}{5.2}=\frac{1}{3}\)
\(\text{hay }\frac{1}{\left[a+b\right]}+\frac{1}{\left[b+c\right]}+\frac{1}{c+a}\le\frac{1}{3}\left(đpcm\right)\)
Đặt \(A=\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+......+\frac{1}{\left(3n-1\right)\left(3n+2\right)}\)
\(=>3A=\frac{3}{2.5}+\frac{3}{5.8}+\frac{3}{8.11}+....+\frac{3}{\left(3n-1\right)\left(3n+2\right)}\)
=> \(3A=\frac{1}{2}-\frac{1}{5}+\frac{1}{5}-\frac{1}{8}+....+\frac{1}{3n-1}-\frac{1}{3n+2}\)
=>\(3A=\frac{1}{2}-\frac{1}{3n+2}\)
=> \(3A=\frac{\left(3n+2\right):2}{3n+2}-\frac{1}{3n+2}\)
=> \(3A=\frac{1,5.n}{3n+2}\)
=>\(A=\frac{1,5.n}{3n+2}.\frac{1}{3}=>A=\frac{1,5.n}{\left(3n+2\right).3}=\frac{1,5.n}{9n+6}\)
\(Hay\) \(A=\frac{1,5n:1,5}{\left(9n+6\right):1,5}=\frac{n}{9n:1,5+6:1,5}=\frac{n}{6n + 4} \left(đpcm\right)\)
Ta có:
\(\frac{1}{a+1}+\frac{1}{a\left(a+1\right)}=\frac{a}{a\left(a+1\right)}+\frac{1}{a\left(a+1\right)}=\frac{a+1}{a\left(a+1\right)}=\frac{1}{a}=y\)
Đúng 100%
\(A=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)...\left(1-\frac{1}{n+1}\right)\)
\(A=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.....\frac{n}{n+1}\)
\(A=\frac{1}{n+1}\)
1)
42n+1+3n+2= (42)n.4 +3n.32
= 16n.4+3n.9
=13n.4+3n.4+3n.9
=13n.4+3n.(4+9)
= 13n.4+3n.13 = 13.(13n-1+3n) chia het cho 13
=> 42n+1+3n+2 chia hết cho 13
2)
\(A=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)...\left(1-\frac{1}{n+1}\right)\)
\(=\frac{1}{2}.\frac{2}{3}....\frac{n}{n+1}\)
\(=\frac{1}{n+1}\)
Đây:
Ta có: \(\frac{1}{a+1}+\frac{1}{a\left(a+1\right)}\)
\(=\frac{a+1}{a\left(a+1\right)}\)
\(=\frac{1}{a}\)
Vậy \(\frac{1}{a+1}+\frac{1}{a\left(a+1\right)=\frac{1}{a}}\)