Calculus Examples

$\sum_{j = 1}^{7} \frac{1}{3} j + 1$

Step 1

Simplify the summation.

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Step 1.1

Combine $\frac{1}{3}$ and $j$ .

$\frac{j}{3} + 1$

Step 1.2

Rewrite the summation.

$\sum_{j = 1}^{7} \frac{j}{3} + 1$

Step 2

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

$\sum_{j = 1}^{7} \frac{j}{3} + 1 = \sum_{j = 1}^{7} \frac{j}{3} + \sum_{j = 1}^{7} 1$

Step 3

Evaluate $\sum_{j = 1}^{7} \frac{j}{3}$ .

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Step 3.1

Factor $\frac{1}{3}$ out of the summation.

$(\frac{1}{3}) \sum_{j = 1}^{7} j$

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Simplify.

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Step 3.4.1

Simplify the expression.

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Step 3.4.1.1

Add $7$ and $1$ .

$\frac{1}{3} \cdot \frac{7 \cdot 8}{2}$

Step 3.4.1.2

Multiply $7$ by $8$ .

$\frac{1}{3} \cdot \frac{56}{2}$

Step 3.4.1.3

Divide $56$ by $2$ .

$\frac{1}{3} \cdot 28$

Step 3.4.2

Combine $\frac{1}{3}$ and $28$ .

$\frac{28}{3}$

Step 4

Evaluate $\sum_{j = 1}^{7} 1$ .

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Step 4.1

The formula for the summation of a constant is:

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

Step 4.2

Substitute the values into the formula.

$(1) (7)$

Step 4.3

Multiply $7$ by $1$ .

$7$

Step 5

Add the results of the summations.

$\frac{28}{3} + 7$

Step 6

Simplify.

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Step 6.1

To write $7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{28}{3} + 7 \cdot \frac{3}{3}$

Step 6.2

Combine $7$ and $\frac{3}{3}$ .

$\frac{28}{3} + \frac{7 \cdot 3}{3}$

Step 6.3

Combine the numerators over the common denominator.

$\frac{28 + 7 \cdot 3}{3}$

Step 6.4

Simplify the numerator.

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Step 6.4.1

Multiply $7$ by $3$ .

$\frac{28 + 21}{3}$

Step 6.4.2

Add $28$ and $21$ .

$\frac{49}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{49}{3}$

Decimal Form:

$16.\overline{3}$

Mixed Number Form:

$16 \frac{1}{3}$