Calculus Examples

$\sum_{i = 0}^{6} 3 i^{2}$

Step 1

Split the summation to make the starting value of $i$ equal to $1$ .

$\sum_{k = 0}^{6} 3 i^{2} = \sum_{k = 1}^{6} 3 i^{2} + \sum_{k = 0}^{0} 3 i^{2}$

Step 2

Evaluate $\sum_{k = 1}^{6} 3 i^{2}$ .

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Step 2.1

Factor $3$ out of the summation.

$3 \sum_{k = 1}^{6} i^{2}$

Step 2.2

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 2.3

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

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

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Add $6$ and $1$ .

$3 \frac{6 \cdot 7 (2 \cdot 6 + 1)}{6}$

Step 2.4.1.2

Combine exponents.

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Step 2.4.1.2.1

Multiply $6$ by $7$ .

$3 \frac{42 (2 \cdot 6 + 1)}{6}$

Step 2.4.1.2.2

Multiply $2$ by $6$ .

$3 \frac{42 (12 + 1)}{6}$

Step 2.4.1.3

Add $12$ and $1$ .

$3 \frac{42 \cdot 13}{6}$

Step 2.4.2

Reduce the expression by cancelling the common factors.

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Step 2.4.2.1

Multiply $42$ by $13$ .

$3 (\frac{546}{6})$

Step 2.4.2.2

Cancel the common factor of $3$ .

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Step 2.4.2.2.1

Factor $3$ out of $6$ .

$3 \frac{546}{3 (2)}$

Step 2.4.2.2.2

Cancel the common factor.

$3 \frac{546}{3 \cdot 2}$

Step 2.4.2.2.3

Rewrite the expression.

$\frac{546}{2}$

Step 2.4.2.3

Divide $546$ by $2$ .

$273$

Step 3

Evaluate $\sum_{k = 0}^{0} 3 i^{2}$ .

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Step 3.1

Expand the series for each value of $i$ .

$3 \cdot 0^{2}$

Step 3.2

Simplify.

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Step 3.2.1

Raise $0$ to the power of $2$ .

$3 \cdot 0$

Step 3.2.2

Multiply $3$ by $0$ .

$0$

Step 4

Replace the summations with the values found.

$273 + 0$

Step 5

Add $273$ and $0$ .

$273$