Basic Math Examples

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

Step 1

To write $\frac{n (n + 1)}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1

Multiply $\frac{n (n + 1)}{2}$ by $\frac{3}{3}$ .

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

Step 2.2

Multiply $2$ by $3$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $n (n + 1)$ out of $n (n + 1) (2 n + 1) + n (n + 1) \cdot 3$ .

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Step 4.1.1

Factor $n (n + 1)$ out of $n (n + 1) \cdot 3$ .

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

Step 4.1.2

Factor $n (n + 1)$ out of $n (n + 1) (2 n + 1) + n (n + 1) (3)$ .

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

Step 4.2

Add $1$ and $3$ .

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

Step 4.3

Factor $2$ out of $2 n + 4$ .

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Step 4.3.1

Factor $2$ out of $2 n$ .

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

Step 4.3.2

Factor $2$ out of $4$ .

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

Step 4.3.3

Factor $2$ out of $2 n + 2 \cdot 2$ .

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

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

Step 5

Cancel the common factor of $2$ and $6$ .

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Step 5.1

Factor $2$ out of $n (n + 1) \cdot 2 (n + 2)$ .

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $2$ out of $6$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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