simplify\:\frac{n^2\left(n+1\right)^2}{4}+\frac{n\left(n+1\right)\left(2n+1\right)}{6}

asked by guest
on Sep 22, 2024 at 1:36 pm



You asked:

Simplify the expression: \(\frac{{n}^{2} \cdot {\left( n + 1 \right)}^{2}}{4} + \frac{n \cdot \left(n + 1\right) \cdot \left(2 n + 1\right)}{6}\)

MathBot Answer:

Simplified:

\[ \frac{{n}^{2} \cdot {\left( n + 1 \right)}^{2}}{4} + \frac{n\left( n + 1 \right) \cdot \left(2 n + 1\right)}{6} = \frac{n^{2} \left(n + 1\right)^{2}}{4} + \frac{n \left(2 n + 1\right) \left(n + 1\right)}{6} \]

Expanded:

\[ \frac{{n}^{2} \cdot {\left( n + 1 \right)}^{2}}{4} + \frac{n\left( n + 1 \right) \cdot \left(2 n + 1\right)}{6} = \frac{n^{4}}{4} + \frac{5 n^{3}}{6} + \frac{3 n^{2}}{4} + \frac{n}{6} \]

Factored:

\[ \frac{{n}^{2} \cdot {\left( n + 1 \right)}^{2}}{4} + \frac{n\left( n + 1 \right) \cdot \left(2 n + 1\right)}{6} = \frac{n \left(3 n + 1\right) \left(n + 1\right) \left(n + 2\right)}{12} \]


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