Calculus Examples

$\sum_{i = 1}^{15} i^{2}$

Step 1

The formula for the summation of a polynomial with degree $2$ is:

$\sum_{k = 1}^{n} i^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

Step 2

Substitute the values into the formula.

$\frac{15 (15 + 1) (2 \cdot 15 + 1)}{6}$

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $15$ and $6$ .

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Step 3.1.1

Factor $3$ out of $15 (15 + 1) (2 \cdot 15 + 1)$ .

$\frac{3 (5 (15 + 1) (2 \cdot 15 + 1))}{6}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $3$ out of $6$ .

$\frac{3 (5 (15 + 1) (2 \cdot 15 + 1))}{3 (2)}$

Step 3.1.2.2

Cancel the common factor.

$\frac{3 (5 (15 + 1) (2 \cdot 15 + 1))}{3 \cdot 2}$

Step 3.1.2.3

Rewrite the expression.

$\frac{5 (15 + 1) (2 \cdot 15 + 1)}{2}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Multiply $2$ by $15$ .

$\frac{5 (15 + 1) (30 + 1)}{2}$

Step 3.2.2

Add $15$ and $1$ .

$\frac{5 \cdot 16 (30 + 1)}{2}$

Step 3.2.3

Multiply $5$ by $16$ .

$\frac{80 (30 + 1)}{2}$

Step 3.2.4

Add $30$ and $1$ .

$\frac{80 \cdot 31}{2}$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $80$ by $31$ .

$\frac{2480}{2}$

Step 3.3.2

Divide $2480$ by $2$ .

$1240$