Solve 3*1/6n(n+1)(2n+1)+4*1/2n(n+1)

$$3 \times \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 ) + 4 \times \frac { 1 } { 2 } n ( n + 1 )$$

Evaluate

$$\frac{n\left(n+1\right)\left(2n+5\right)}{2}$$

Solution Steps

Multiply 3 and \frac{1}{6} to get \frac{3}{6}.

$$\frac{3}{6}n\left(n+1\right)\left(2n+1\right)+4\times \frac{1}{2}n\left(n+1\right)$$

Reduce the fraction \frac{3}{6} to lowest terms by extracting and canceling out 3.

$$\frac{1}{2}n\left(n+1\right)\left(2n+1\right)+4\times \frac{1}{2}n\left(n+1\right)$$

Use the distributive property to multiply \frac{1}{2}n by n+1.

$$\left(\frac{1}{2}nn+\frac{1}{2}n\right)\left(2n+1\right)+4\times \frac{1}{2}n\left(n+1\right)$$

Multiply n and n to get n^{2}.

$$\left(\frac{1}{2}n^{2}+\frac{1}{2}n\right)\left(2n+1\right)+4\times \frac{1}{2}n\left(n+1\right)$$

Apply the distributive property by multiplying each term of \frac{1}{2}n^{2}+\frac{1}{2}n by each term of 2n+1.

$$\frac{1}{2}n^{2}\times 2n+\frac{1}{2}n^{2}+\frac{1}{2}n\times 2n+\frac{1}{2}n+4\times \frac{1}{2}n\left(n+1\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{1}{2}n^{3}\times 2+\frac{1}{2}n^{2}+\frac{1}{2}n\times 2n+\frac{1}{2}n+4\times \frac{1}{2}n\left(n+1\right)$$

Multiply n and n to get n^{2}.

$$\frac{1}{2}n^{3}\times 2+\frac{1}{2}n^{2}+\frac{1}{2}n^{2}\times 2+\frac{1}{2}n+4\times \frac{1}{2}n\left(n+1\right)$$

Cancel out 2 and 2.

$$n^{3}+\frac{1}{2}n^{2}+\frac{1}{2}n^{2}\times 2+\frac{1}{2}n+4\times \frac{1}{2}n\left(n+1\right)$$

Cancel out 2 and 2.

$$n^{3}+\frac{1}{2}n^{2}+n^{2}+\frac{1}{2}n+4\times \frac{1}{2}n\left(n+1\right)$$

Combine \frac{1}{2}n^{2} and n^{2} to get \frac{3}{2}n^{2}.

$$n^{3}+\frac{3}{2}n^{2}+\frac{1}{2}n+4\times \frac{1}{2}n\left(n+1\right)$$

Multiply 4 and \frac{1}{2} to get \frac{4}{2}.

$$n^{3}+\frac{3}{2}n^{2}+\frac{1}{2}n+\frac{4}{2}n\left(n+1\right)$$

Divide 4 by 2 to get 2.

$$n^{3}+\frac{3}{2}n^{2}+\frac{1}{2}n+2n\left(n+1\right)$$

Use the distributive property to multiply 2n by n+1.

$$n^{3}+\frac{3}{2}n^{2}+\frac{1}{2}n+2n^{2}+2n$$

Combine \frac{3}{2}n^{2} and 2n^{2} to get \frac{7}{2}n^{2}.

$$n^{3}+\frac{7}{2}n^{2}+\frac{1}{2}n+2n$$

Combine \frac{1}{2}n and 2n to get \frac{5}{2}n.

$$n^{3}+\frac{7}{2}n^{2}+\frac{5}{2}n$$

Expand

$$n^{3}+\frac{7n^{2}}{2}+\frac{5n}{2}$$

Solution Steps

Multiply 3 and \frac{1}{6} to get \frac{3}{6}.

$$\frac{3}{6}n\left(n+1\right)\left(2n+1\right)+4\times \frac{1}{2}n\left(n+1\right)$$

Reduce the fraction \frac{3}{6} to lowest terms by extracting and canceling out 3.

$$\frac{1}{2}n\left(n+1\right)\left(2n+1\right)+4\times \frac{1}{2}n\left(n+1\right)$$

Use the distributive property to multiply \frac{1}{2}n by n+1.

$$\left(\frac{1}{2}nn+\frac{1}{2}n\right)\left(2n+1\right)+4\times \frac{1}{2}n\left(n+1\right)$$

Multiply n and n to get n^{2}.

$$\left(\frac{1}{2}n^{2}+\frac{1}{2}n\right)\left(2n+1\right)+4\times \frac{1}{2}n\left(n+1\right)$$

Apply the distributive property by multiplying each term of \frac{1}{2}n^{2}+\frac{1}{2}n by each term of 2n+1.

$$\frac{1}{2}n^{2}\times 2n+\frac{1}{2}n^{2}+\frac{1}{2}n\times 2n+\frac{1}{2}n+4\times \frac{1}{2}n\left(n+1\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{1}{2}n^{3}\times 2+\frac{1}{2}n^{2}+\frac{1}{2}n\times 2n+\frac{1}{2}n+4\times \frac{1}{2}n\left(n+1\right)$$

Multiply n and n to get n^{2}.

$$\frac{1}{2}n^{3}\times 2+\frac{1}{2}n^{2}+\frac{1}{2}n^{2}\times 2+\frac{1}{2}n+4\times \frac{1}{2}n\left(n+1\right)$$

Cancel out 2 and 2.

$$n^{3}+\frac{1}{2}n^{2}+\frac{1}{2}n^{2}\times 2+\frac{1}{2}n+4\times \frac{1}{2}n\left(n+1\right)$$

Cancel out 2 and 2.

$$n^{3}+\frac{1}{2}n^{2}+n^{2}+\frac{1}{2}n+4\times \frac{1}{2}n\left(n+1\right)$$

Combine \frac{1}{2}n^{2} and n^{2} to get \frac{3}{2}n^{2}.

$$n^{3}+\frac{3}{2}n^{2}+\frac{1}{2}n+4\times \frac{1}{2}n\left(n+1\right)$$

Multiply 4 and \frac{1}{2} to get \frac{4}{2}.

$$n^{3}+\frac{3}{2}n^{2}+\frac{1}{2}n+\frac{4}{2}n\left(n+1\right)$$

Divide 4 by 2 to get 2.

$$n^{3}+\frac{3}{2}n^{2}+\frac{1}{2}n+2n\left(n+1\right)$$

Use the distributive property to multiply 2n by n+1.

$$n^{3}+\frac{3}{2}n^{2}+\frac{1}{2}n+2n^{2}+2n$$

Combine \frac{3}{2}n^{2} and 2n^{2} to get \frac{7}{2}n^{2}.

$$n^{3}+\frac{7}{2}n^{2}+\frac{1}{2}n+2n$$

Combine \frac{1}{2}n and 2n to get \frac{5}{2}n.

$$n^{3}+\frac{7}{2}n^{2}+\frac{5}{2}n$$