Calculus Examples

$\sum_{i = 3}^{13} {(i - 3)}^{2}$

Step 1

Simplify the summation.

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Step 1.1

Rewrite ${(i - 3)}^{2}$ as $(i - 3) (i - 3)$ .

$(i - 3) (i - 3)$

Step 1.2

Expand $(i - 3) (i - 3)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$i (i - 3) - 3 (i - 3)$

Step 1.2.2

Apply the distributive property.

$i \cdot i + i \cdot - 3 - 3 (i - 3)$

Step 1.2.3

Apply the distributive property.

$i \cdot i + i \cdot - 3 - 3 i - 3 \cdot - 3$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $i$ by $i$ .

$i^{2} + i \cdot - 3 - 3 i - 3 \cdot - 3$

Step 1.3.1.2

Move $- 3$ to the left of $i$ .

$i^{2} - 3 \cdot i - 3 i - 3 \cdot - 3$

Step 1.3.1.3

Multiply $- 3$ by $- 3$ .

$i^{2} - 3 i - 3 i + 9$

Step 1.3.2

Subtract $3 i$ from $- 3 i$ .

$i^{2} - 6 i + 9$

Step 1.4

Rewrite the summation.

$\sum_{i = 3}^{13} i^{2} - 6 i + 9$

Step 2

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

$\sum_{i = 3}^{13} i^{2} - 6 i + 9 = \sum_{i = 1}^{13} i^{2} - 6 i + 9 - \sum_{i = 1}^{2} i^{2} - 6 i + 9$

Step 3

Evaluate $\sum_{i = 1}^{13} i^{2} - 6 i + 9$ .

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Step 3.1

Split the summation into smaller summations that fit the summation rules.

$\sum_{k = 1}^{13} i^{2} - 6 i + 9 = \sum_{k = 1}^{13} i^{2} - 6 \sum_{k = 1}^{13} i + \sum_{k = 1}^{13} 9$

Step 3.2

Evaluate $\sum_{k = 1}^{13} i^{2}$ .

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Step 3.2.1

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 3.2.2

Substitute the values into the formula.

$\frac{13 (13 + 1) (2 \cdot 13 + 1)}{6}$

Step 3.2.3

Simplify.

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Step 3.2.3.1

Simplify the numerator.

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Step 3.2.3.1.1

Add $13$ and $1$ .

$\frac{13 \cdot 14 (2 \cdot 13 + 1)}{6}$

Step 3.2.3.1.2

Combine exponents.

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Step 3.2.3.1.2.1

Multiply $13$ by $14$ .

$\frac{182 (2 \cdot 13 + 1)}{6}$

Step 3.2.3.1.2.2

Multiply $2$ by $13$ .

$\frac{182 (26 + 1)}{6}$

Step 3.2.3.1.3

Add $26$ and $1$ .

$\frac{182 \cdot 27}{6}$

Step 3.2.3.2

Simplify the expression.

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Step 3.2.3.2.1

Multiply $182$ by $27$ .

$\frac{4914}{6}$

Step 3.2.3.2.2

Divide $4914$ by $6$ .

$819$

Step 3.3

Evaluate $6 \sum_{k = 1}^{13} i$ .

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Step 3.3.1

The formula for the summation of a polynomial with degree $1$ is:

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

Step 3.3.2

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

$(- 6) (\frac{13 (13 + 1)}{2})$

Step 3.3.3

Simplify.

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Step 3.3.3.1

Simplify the expression.

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Step 3.3.3.1.1

Add $13$ and $1$ .

$- 6 \frac{13 \cdot 14}{2}$

Step 3.3.3.1.2

Multiply $13$ by $14$ .

$- 6 (\frac{182}{2})$

Step 3.3.3.2

Cancel the common factor of $2$ .

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Step 3.3.3.2.1

Factor $2$ out of $- 6$ .

$2 (- 3) \frac{182}{2}$

Step 3.3.3.2.2

Cancel the common factor.

$2 \cdot - 3 \frac{182}{2}$

Step 3.3.3.2.3

Rewrite the expression.

$- 3 \cdot 182$

Step 3.3.3.3

Multiply $- 3$ by $182$ .

$- 546$

Step 3.4

Evaluate $\sum_{k = 1}^{13} 9$ .

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Step 3.4.1

The formula for the summation of a constant is:

$\sum_{k = 1}^{n} c = c n$

Step 3.4.2

Substitute the values into the formula.

$(9) (13)$

Step 3.4.3

Multiply $9$ by $13$ .

$117$

Step 3.5

Add the results of the summations.

$819 - 546 + 117$

Step 3.6

Simplify.

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Step 3.6.1

Subtract $546$ from $819$ .

$273 + 117$

Step 3.6.2

Add $273$ and $117$ .

$390$

Step 4

Evaluate $\sum_{i = 1}^{2} i^{2} - 6 i + 9$ .

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Step 4.1

Expand the series for each value of $i$ .

$1^{2} - 6 \cdot 1 + 9 + 2^{2} - 6 \cdot 2 + 9$

Step 4.2

Simplify.

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Step 4.2.1

Raise $1$ to the power of $2$ .

$1 - 6 \cdot 1 + 9 + 2^{2} - 6 \cdot 2 + 9$

Step 4.2.2

Multiply $- 6$ by $1$ .

$1 - 6 + 9 + 2^{2} - 6 \cdot 2 + 9$

Step 4.2.3

Subtract $6$ from $1$ .

$- 5 + 9 + 2^{2} - 6 \cdot 2 + 9$

Step 4.2.4

Add $- 5$ and $9$ .

$4 + 2^{2} - 6 \cdot 2 + 9$

Step 4.2.5

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

$4 + 4 - 6 \cdot 2 + 9$

Step 4.2.6

Multiply $- 6$ by $2$ .

$4 + 4 - 12 + 9$

Step 4.2.7

Subtract $12$ from $4$ .

$4 - 8 + 9$

Step 4.2.8

Add $- 8$ and $9$ .

$4 + 1$

Step 4.2.9

Add $4$ and $1$ .

$5$

Step 5

Replace the summations with the values found.

$390 - 5$

Step 6

Subtract $5$ from $390$ .

$385$