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\(a.\) \(\frac{6^3+3.6^2+3^3}{-13}=\frac{2^3.3^3+3.3^2.2^2+3^3}{-13}=\frac{2^3.3^3+3^3.2^2+3^3}{-13}\)
\(=\frac{3^3.\left(2^3+2^2+1\right)}{-13}=\frac{3^3.13}{-13}=\frac{3^3.\left(-1\right)}{1}=-27\)
\(b.\)\(A=2^2+4^2+6^2+...+20^2=2^2\left(1+2^2+3^2+...+10^2\right)\)
\(A=2^2.\frac{10.\left(10+1\right).\left(2.10+1\right)}{6}=4.385=1540\)
( Ta có: công thức tính tổng bình phương liên tiếp tứ 1 đến n là: \(1^2+2^2+3^2+...+n^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}\))
\(c.\)\(B=100^2+200^2+...+1000^2=\left(100.1\right)^2+\left(100.2\right)^2+...+\left(100.10\right)^2\)
\(B=100^2.1^2+100^2.2^2+...+100^2.10^2=100^2.\left(1^2+2^2+...+10^2\right)\)
Áp dụng công thức \(1^2+2^2+3^2+...+n^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}\)
Ta có: \(B=100^2\times385=3,850,000\)
\(\left(\frac{1}{2^2}-1\right)\cdot\left(\frac{1}{3^2}-1\right)\cdot..\cdot\left(\frac{1}{10^2}-1\right)\)
\(=\left(\frac{1}{2}\cdot\frac{1}{2}-1\right)\cdot\left(\frac{1}{3}\cdot\frac{1}{3}-1\right)\cdot...\cdot\left(\frac{1}{10}\cdot\frac{1}{10}-1\right)\)
\(=\left(\frac{1}{4}-1\right)\cdot\left(\frac{1}{9}-1\right)\cdot...\cdot\left(\frac{1}{100}-1\right)\)
\(=\frac{-3}{4}\cdot\frac{-8}{9}\cdot...\cdot\frac{-99}{100}\)
\(=\frac{\left(-1\right).\left(-3\right)}{2.2}\cdot\frac{\left(-2\right).\left(-4\right)}{3.3}\cdot...\cdot\frac{\left(-9\right).\left(-11\right)}{10.10}\)
\(=\frac{\left(-1\right).\left(-2\right)....\left(-9\right)}{2.3....10}\cdot\frac{\left(-3\right).\left(-4\right)....\left(-11\right)}{2.3.....10}\)
\(=\frac{-1}{10}\cdot\frac{-11}{2}=\frac{-11}{20}\)
12+22+32+..........+20132+20142+20152 Gọi dãy trên là A
=1x1+2x2+3x3+.........+2013x2013+2014x2014+2015x2015
=1x(2-1)+2x(3-1)+3x(4-1)+........+2013x(2014-1)+2014x(2015-1)+2015x(2016-1)
=1x2-1x1+2x3-2x1+3x4-3x1+......+2013x2014-2013x1+2014x2015-2014x1+2015x2016-2015x1
=(1x2+2x3+3x4+.........+2013x2014+2014x2015+2015x2016)-(1+2+3+........+2013+2014+2015)
Gọi vế 1 của dãy là a
3xa=1x2x3+2x3x(4-1)+3x4x(5-2)+......+2013x2014x(2015-2012)+2014x2015x(2016-2013)+2015x2016x(2017-2014)
3xa=1x2x3+2x3x4-1x2x3+3x4x5-2x3x4+........+2013x2014x2015-2012x2013x2014+2014x2015x2016-2013x2014x2015+2015x2016x2017-2014x2015x2016
a=2015x2016x2017:3
a=2731179360
A=2731179360-(1+2+3+.....+2013+2014+2015)
A=2731179360-[2015x(2015+1):2]
A=2731179360-2031120
A=2729143240
Nhớ tick cho mình nha
A = 1+22+23+24+......+22013
2A = 22+23+24+......+22013+22014
=> 2A - A = A = 22015 - 1
Vậy A = 22015 - 1
\(D=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+.......+\dfrac{1}{10^2}\)
\(D< \dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+.......+\dfrac{1}{9.10}\)
\(D< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+.....+\dfrac{1}{9}-\dfrac{1}{10}\)
\(D< 1-\dfrac{1}{10}\Leftrightarrow D< 1\left(đpcm\right)\)