Calculus Examples

$\sum_{i = 1}^{5} i^{2}$

Step 1

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

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

Step 2

Substitute the values into the formula and make sure to multiply by the front term.

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

Step 3

Simplify.

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Step 3.1

Remove parentheses.

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

Add $5$ and $1$ .

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

Step 3.2.2

Combine exponents.

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Step 3.2.2.1

Multiply $5$ by $6$ .

$\frac{30 (2 \cdot 5 + 1)}{6}$

Step 3.2.2.2

Multiply $2$ by $5$ .

$\frac{30 (10 + 1)}{6}$

Step 3.2.3

Add $10$ and $1$ .

$\frac{30 \cdot 11}{6}$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $30$ by $11$ .

$\frac{330}{6}$

Step 3.3.2

Divide $330$ by $6$ .

$55$