Calculus Examples

$\sum_{i = 5}^{13} 3 i + 2$

Step 1

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

$\sum_{k = 5}^{13} 3 i + 2 = \sum_{k = 1}^{13} 3 i + 2 - \sum_{k = 1}^{4} 3 i + 2$

Step 2

Evaluate $\sum_{k = 1}^{13} 3 i + 2$ .

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Step 2.1

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

$\sum_{k = 1}^{13} 3 i + 2 = 3 \sum_{k = 1}^{13} i + \sum_{k = 1}^{13} 2$

Step 2.2

Evaluate $3 \sum_{k = 1}^{13} i$ .

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Step 2.2.1

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

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

Step 2.2.2

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

$(3) (\frac{13 (13 + 1)}{2})$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Add $13$ and $1$ .

$3 \frac{13 \cdot 14}{2}$

Step 2.2.3.2

Multiply $13$ by $14$ .

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

Step 2.2.3.3

Divide $182$ by $2$ .

$3 \cdot 91$

Step 2.2.3.4

Multiply $3$ by $91$ .

$273$

Step 2.3

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

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Step 2.3.1

The formula for the summation of a constant is:

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

Step 2.3.2

Substitute the values into the formula.

$(2) (13)$

Step 2.3.3

Multiply $2$ by $13$ .

$26$

Step 2.4

Add the results of the summations.

$273 + 26$

Step 2.5

Add $273$ and $26$ .

$299$

Step 3

Evaluate $\sum_{k = 1}^{4} 3 i + 2$ .

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Step 3.1

Expand the series for each value of $i$ .

$3 \cdot 1 + 2 + 3 \cdot 2 + 2 + 3 \cdot 3 + 2 + 3 \cdot 4 + 2$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $3$ by $1$ .

$3 + 2 + 3 \cdot 2 + 2 + 3 \cdot 3 + 2 + 3 \cdot 4 + 2$

Step 3.2.2

Add $3$ and $2$ .

$5 + 3 \cdot 2 + 2 + 3 \cdot 3 + 2 + 3 \cdot 4 + 2$

Step 3.2.3

Multiply $3$ by $2$ .

$5 + 6 + 2 + 3 \cdot 3 + 2 + 3 \cdot 4 + 2$

Step 3.2.4

Add $6$ and $2$ .

$5 + 8 + 3 \cdot 3 + 2 + 3 \cdot 4 + 2$

Step 3.2.5

Add $5$ and $8$ .

$13 + 3 \cdot 3 + 2 + 3 \cdot 4 + 2$

Step 3.2.6

Multiply $3$ by $3$ .

$13 + 9 + 2 + 3 \cdot 4 + 2$

Step 3.2.7

Add $9$ and $2$ .

$13 + 11 + 3 \cdot 4 + 2$

Step 3.2.8

Add $13$ and $11$ .

$24 + 3 \cdot 4 + 2$

Step 3.2.9

Multiply $3$ by $4$ .

$24 + 12 + 2$

Step 3.2.10

Add $12$ and $2$ .

$24 + 14$

Step 3.2.11

Add $24$ and $14$ .

$38$

Step 4

Replace the summations with the values found.

$299 - 38$

Step 5

Subtract $38$ from $299$ .

$261$