Calculus Examples

$\sum_{n = 1}^{34} \frac{3 n}{4}$

Step 1

Factor $\frac{3}{4}$ out of the summation.

$(\frac{3}{4}) \sum_{n = 1}^{34} n$

Step 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

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

$(\frac{3}{4}) (\frac{34 (34 + 1)}{2})$

Step 4

Simplify.

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Step 4.1

Simplify the expression.

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Step 4.1.1

Add $34$ and $1$ .

$\frac{3}{4} \cdot \frac{34 \cdot 35}{2}$

Step 4.1.2

Multiply $34$ by $35$ .

$\frac{3}{4} \cdot \frac{1190}{2}$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Factor $2$ out of $4$ .

$\frac{3}{2 (2)} \cdot \frac{1190}{2}$

Step 4.2.2

Factor $2$ out of $1190$ .

$\frac{3}{2 \cdot 2} \cdot \frac{2 \cdot 595}{2}$

Step 4.2.3

Cancel the common factor.

$\frac{3}{2 \cdot 2} \cdot \frac{2 \cdot 595}{2}$

Step 4.2.4

Rewrite the expression.

$\frac{3}{2} \cdot \frac{595}{2}$

Step 4.3

Multiply $\frac{3}{2}$ by $\frac{595}{2}$ .

$\frac{3 \cdot 595}{2 \cdot 2}$

Step 4.4

Multiply.

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Step 4.4.1

Multiply $3$ by $595$ .

$\frac{1785}{2 \cdot 2}$

Step 4.4.2

Multiply $2$ by $2$ .

$\frac{1785}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1785}{4}$

Decimal Form:

$446.25$

Mixed Number Form:

$446 \frac{1}{4}$