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受欢迎的代数 >

展开 (n(n+1)(2n+1))/6+3((n(n+1))/2)

解答

展开

\frac{n ( n + 1 ) ( 2 n + 1 )}{6} + 3 (\frac{n ( n + 1 )}{2})

解答

\frac{n ^{3}}{3} + 2 n^{2} + \frac{5 n}{3}

求解步骤

\frac{n ( n + 1 ) ( 2 n + 1 )}{6} + 3 (\frac{n ( n + 1 )}{2})

去除括号:

(a) = a

= \frac{n ( n + 1 ) ( 2 n + 1 )}{6} + 3 \cdot \frac{n ( n + 1 )}{2}

\frac{n ( n + 1 ) ( 2 n + 1 )}{6} = \frac{2 n ^{3} + 3 n ^{2} + n}{6}

3 \cdot \frac{n ( n + 1 )}{2} = \frac{3 n ^{2} + 3 n}{2}

= \frac{2 n ^{3} + 3 n ^{2} + n}{6} + \frac{3 n ^{2} + 3 n}{2}

6, 2

的最小公倍数:

6

根据最小公倍数调整分式

= \frac{2 n ^{3} + 3 n ^{2} + n}{6} + \frac{( 3 n ^{2} + 3 n ) \cdot 3}{6}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 n ^{3} + 3 n ^{2} + n + ( 3 n ^{2} + 3 n ) \cdot 3}{6}

分解

2 n^{3} + 3 n^{2} + n + (3 n^{2} + 3 n) 3 : n \cdot 2 (n^{2} + 6 n + 5)

= \frac{n \cdot 2 ( n ^{2} + 6 n + 5 )}{6}

约分:

2

= \frac{n ( n ^{2} + 6 n + 5 )}{3}

乘开

n (n^{2} + 6 n + 5) : n^{3} + 6 n^{2} + 5 n

= \frac{n ^{3} + 6 n ^{2} + 5 n}{3}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{n ^{3} + 6 n ^{2} + 5 n}{3} = \frac{n ^{3}}{3} + \frac{6 n ^{2}}{3} + \frac{5 n}{3}

= \frac{n ^{3}}{3} + \frac{6 n ^{2}}{3} + \frac{5 n}{3}

数字相除:

\frac{6}{3} = 2

= \frac{n ^{3}}{3} + 2 n^{2} + \frac{5 n}{3}

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