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Ta thấy: 1+ 2/ n^2+3n = n^2+3n+2 / n(n+3) =(n+1)(n+2) /n(n+3)
Áp dụng công thức trên,ta có:
A= (1+2/4 )(1+ 2/10)(1+2/18).....(1+2/ n^2+3n)
=(1+2 /1x4)( 1+2 /2x5)(1+2 /3x6).....[ (n+1)(n+2)/ n(n+3)]
=(2x3 /1x4)(3x4 /2x5)(4x5 /3x6).....[ (n+1)(n+2) /n(n+3)]
= 3x(n+1 /n+3)
Vì n+1 /n+3 <1 với mọi n thuộc N nên 3x(n+1 /n+3) <3
Vậy A<3
\(\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)...\left(1-\frac{1}{n+1}\right)\)
\(=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}...\frac{n}{n+1}\)
\(=\frac{1}{n+1}\)
\(1+\frac{1}{2}.\left(1+2\right)+\frac{1}{3}.\left(1+2+3\right)...+\frac{1}{20}.\left(1+2+3+...+20\right)\)
\(=1+\frac{1}{2}.2.3:2+\frac{1}{3}.3.4:2+\frac{1}{4}.4.5:2+...+\frac{1}{20}.20.21:2\)
\(=\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+\frac{5}{2}+...+\frac{21}{2}\)
\(=\frac{2+3+4+5+...+21}{2}=115\)
Bài đầu đơn giản rồi , tự tính nhé <3
Bài 2
\(3^{n+2}-2^{n+2}+3^n-2^n\)
\(=3^n.3^2-2^n.2^2+3^n-2^n\)
\(=\left(3^n.3^2+1\right)-\left(2^n.2^2+1\right)\)
\(=3^n.10-2^n.5\)
\(=3^n.10-2^{n-1}.10\)
\(=10.\left(3^n-2^{n-1}\right)⋮10\)
Vậy.....
Lời giải:
\(A=\frac{2-1}{2}.\frac{3-1}{3}.\frac{4-1}{4}.....\frac{n+1-1}{n+1}=\frac{1.2.3....n}{2.3.4...n(n+1)}=\frac{1}{n+1}\)
\(A=\left(\frac{1}{2}-1\right)\left(\frac{1}{3}-1\right)...\left(\frac{1}{2002}-1\right)\left(\frac{1}{2003}-1\right)\)
\(=\left(-\frac{1}{2}\right)\left(-\frac{2}{3}\right)...\left(-\frac{2001}{2002}\right)\left(-\frac{2002}{2003}\right)\)
\(=\frac{-1.\left(-2\right).....\left(-2001\right)\left(-2002\right)}{2.3....2002.2003}\)
\(=\frac{1}{2003}\)
\(\frac{1}{\left(n-1\right)n\left(n+1\right)}=\frac{1}{2}.\frac{2}{\left(n-1\right)n\left(n+1\right)}=\frac{1}{2}.\frac{\left(n+1\right)-\left(n-1\right)}{\left(n-1\right)n\left(n+1\right)}\)
\(=\frac{1}{2}\left[\frac{1}{\left(n-1\right)n}-\frac{1}{n\left(n+1\right)}\right]\)
Ta có đpcm.
3n+2 - 2n+2 +3n - 2n = 3n . 32 - 2n. 22 +3n -2n
= 3n(32+1) - (2n.22 +2n)
=3n . 10 - 2n .5
=3n.10 - 2n-1 .2 .5
= 3n.10 - 2n-1 .10
= 10(3n - 2n-1)
vì 10 chia hết cho 10 nên 10(3n-2n-1) chia hết cho 10
=> 3n+2 - 2n+2 +3n -2n chia hết cho 10
Ai làm nhanh nhất mình sẽ **** xin cảm ơn các bạn mình đang cần gấp
Xét n trong các trường hợp sau:
+) n = 4k (k \(\in\) N) => VT = \(\left[\frac{4k+3}{4}\right]+\left[\frac{4k+5}{4}\right]+\left[\frac{4k}{2}\right]=\left[k+0,75\right]+\left[k+1,25\right]+\left[2k\right]\)
\(=k+\left(k+1\right)+2k=4k+1=n+1\)= VP
+) n = 4k + 1 (k \(\in\) N) => VT = \(\left[\frac{4k+4}{4}\right]+\left[\frac{4k+6}{4}\right]+\left[\frac{4k+1}{2}\right]=\left[k+1\right]+\left[k+1,5\right]+\left[2k+0,5\right]\)
\(=\left(k+1\right)+\left(k+1\right)+2k=4k+2=n+1\)= VP
+) n = 4k + 2 (k \(\in\) N) => VT= \(\left[\frac{4k+5}{4}\right]+\left[\frac{4k+7}{4}\right]+\left[\frac{4k+2}{2}\right]=\left[k+1,25\right]+\left[k+1,75\right]+\left[2k+1\right]\)
\(=\left(k+1\right)+\left(k+1\right)+\left(2k+1\right)=4k+3=n+1\)= VP
+) n = 4k + 3 (k \(\in\) N) => VT = \(\left[\frac{4k+6}{4}\right]+\left[\frac{4k+8}{4}\right]+\left[\frac{4k+3}{2}\right]=\left[k+1,5\right]+\left[k+2\right]+\left[2k+1,5\right]\)
\(=\left(k+1\right)+\left(k+2\right)+\left(2k+1\right)=4k+4=n+1\)= VP
Từ các trường hợp trên => đpcm
\(\frac{n+3}{4}+\frac{n+5}{4}+\frac{n}{2}=\frac{n+3}{4}+\frac{n+5}{4}+\frac{2n}{4}=\frac{n+3+n+5+2n}{4}=\frac{4n+8}{4}=n+2\)