What is expand (((n(n+1)))/2)^2 ?

The solution to expand (((n(n+1)))/2)^2 is (n^4)/4+(n^3)/2+(n^2)/4

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expand (((n(n+1)))/2)^2

expand

(\frac{( n ( n + 1 ) )}{2})^{2}

\frac{n ^{4}}{4} + \frac{n ^{3}}{2} + \frac{n ^{2}}{4}

Solution steps

(\frac{n ( n + 1 )}{2})^{2}

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

= (\frac{n ^{2} + n}{2})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( n ^{2} + n ) ^{2}}{2 ^{2}}

(n^{2} + n)^{2} = n^{4} + 2 n^{3} + n^{2}

= \frac{n ^{4} + 2 n ^{3} + n ^{2}}{2 ^{2}}

2^{2} = 4

= \frac{n ^{4} + 2 n ^{3} + n ^{2}}{4}

Apply the fraction rule:

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

\frac{n ^{4} + 2 n ^{3} + n ^{2}}{4} = \frac{n ^{4}}{4} + \frac{2 n ^{3}}{4} + \frac{n ^{2}}{4}

= \frac{n ^{4}}{4} + \frac{2 n ^{3}}{4} + \frac{n ^{2}}{4}

Cancel

\frac{2 n ^{3}}{4} : \frac{n ^{3}}{2}

= \frac{n ^{4}}{4} + \frac{n ^{3}}{2} + \frac{n ^{2}}{4}

cos (\frac{1}{n ^{2}}) - cos (\frac{1}{( n + 1 ) ^{2}})

\frac{x + 4}{( x + 2 ) ( x - 3 )}

- \frac{k + 1}{k ( k - 1 )}

\frac{2 e ^{- 2 s}}{s ^{2} ( 4 s + 1 )}

\frac{3 x + 2 + ( 7 x - 1 ) 2}{4}

What is expand (((n(n+1)))/2)^2 ?
The solution to expand (((n(n+1)))/2)^2 is (n^4)/4+(n^3)/2+(n^2)/4