Precalculus Examples

$\sum_{k = 1}^{25} k^{2}$

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.

$\frac{25 (25 + 1) (2 \cdot 25 + 1)}{6}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $25$ and $1$ .

$\frac{25 \cdot 26 (2 \cdot 25 + 1)}{6}$

Step 3.1.2

Combine exponents.

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Step 3.1.2.1

Multiply $25$ by $26$ .

$\frac{650 (2 \cdot 25 + 1)}{6}$

Step 3.1.2.2

Multiply $2$ by $25$ .

$\frac{650 (50 + 1)}{6}$

Step 3.1.3

Add $50$ and $1$ .

$\frac{650 \cdot 51}{6}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Multiply $650$ by $51$ .

$\frac{33150}{6}$

Step 3.2.2

Divide $33150$ by $6$ .

$5525$