Calculus Examples

$\sum_{i = 6}^{14} 9 i^{2}$

Step 1

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

$\sum_{k = 6}^{14} 9 i^{2} = \sum_{k = 1}^{14} 9 i^{2} - \sum_{k = 1}^{5} 9 i^{2}$

Step 2

Evaluate $\sum_{k = 1}^{14} 9 i^{2}$ .

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Step 2.1

Factor $9$ out of the summation.

$9 \sum_{k = 1}^{14} i^{2}$

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.

$(9) (\frac{14 (14 + 1) (2 \cdot 14 + 1)}{6})$

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 $14$ and $1$ .

$9 \frac{14 \cdot 15 (2 \cdot 14 + 1)}{6}$

Step 2.4.1.2

Combine exponents.

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Step 2.4.1.2.1

Multiply $14$ by $15$ .

$9 \frac{210 (2 \cdot 14 + 1)}{6}$

Step 2.4.1.2.2

Multiply $2$ by $14$ .

$9 \frac{210 (28 + 1)}{6}$

Step 2.4.1.3

Add $28$ and $1$ .

$9 \frac{210 \cdot 29}{6}$

Step 2.4.2

Simplify terms.

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Step 2.4.2.1

Multiply $210$ by $29$ .

$9 (\frac{6090}{6})$

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 $9$ .

$3 (3) \frac{6090}{6}$

Step 2.4.2.2.2

Factor $3$ out of $6$ .

$3 \cdot 3 \frac{6090}{3 \cdot 2}$

Step 2.4.2.2.3

Cancel the common factor.

$3 \cdot 3 \frac{6090}{3 \cdot 2}$

Step 2.4.2.2.4

Rewrite the expression.

$3 (\frac{6090}{2})$

Step 2.4.2.3

Combine $3$ and $\frac{6090}{2}$ .

$\frac{3 \cdot 6090}{2}$

Step 2.4.2.4

Simplify the expression.

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Step 2.4.2.4.1

Multiply $3$ by $6090$ .

$\frac{18270}{2}$

Step 2.4.2.4.2

Divide $18270$ by $2$ .

$9135$

Step 3

Evaluate $\sum_{k = 1}^{5} 9 i^{2}$ .

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Step 3.1

Expand the series for each value of $i$ .

$9 \cdot 1^{2} + 9 \cdot 2^{2} + 9 \cdot 3^{2} + \dots + 9 \cdot 5^{2}$

Step 3.2

Simplify the expanded form.

$495$

Step 4

Replace the summations with the values found.

$9135 - 495$

Step 5

Subtract $495$ from $9135$ .

$8640$