Calculus Examples

$\sum_{i = 3}^{10} \sqrt{3} \cdot \frac{i}{2}$

Step 1

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

$\sum_{i = 3}^{10} \frac{\sqrt{3} i}{2} = \sum_{i = 1}^{10} \frac{\sqrt{3} i}{2} - \sum_{i = 1}^{2} \frac{\sqrt{3} i}{2}$

Step 2

Evaluate $\sum_{i = 1}^{10} \frac{\sqrt{3} i}{2}$ .

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Step 2.1

Factor $\frac{\sqrt{3}}{2}$ out of the summation.

$\frac{\sqrt{3}}{2} \sum_{k = 1}^{10} i$

Step 2.2

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

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

Step 2.3

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

$(\frac{\sqrt{3}}{2}) (\frac{10 (10 + 1)}{2})$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the expression.

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Step 2.4.1.1

Add $10$ and $1$ .

$\frac{\sqrt{3}}{2} \cdot \frac{10 \cdot 11}{2}$

Step 2.4.1.2

Multiply $10$ by $11$ .

$\frac{\sqrt{3}}{2} \cdot \frac{110}{2}$

Step 2.4.2

Cancel the common factor of $2$ .

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Step 2.4.2.1

Factor $2$ out of $110$ .

$\frac{\sqrt{3}}{2} \cdot \frac{2 (55)}{2}$

Step 2.4.2.2

Cancel the common factor.

$\frac{\sqrt{3}}{2} \cdot \frac{2 \cdot 55}{2}$

Step 2.4.2.3

Rewrite the expression.

$\sqrt{3} \frac{55}{2}$

Step 2.4.3

Combine $\sqrt{3}$ and $\frac{55}{2}$ .

$\frac{\sqrt{3} \cdot 55}{2}$

Step 2.4.4

Move $55$ to the left of $\sqrt{3}$ .

$\frac{55 \sqrt{3}}{2}$

Step 3

Evaluate $\sum_{i = 1}^{2} \frac{\sqrt{3} i}{2}$ .

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Step 3.1

Expand the series for each value of $i$ .

$\frac{\sqrt{3} \cdot 1}{2} + \frac{\sqrt{3} \cdot 2}{2}$

Step 3.2

Simplify.

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Step 3.2.1

Combine the numerators over the common denominator.

$\frac{\sqrt{3} \cdot 1 + \sqrt{3} \cdot 2}{2}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Multiply $\sqrt{3}$ by $1$ .

$\frac{\sqrt{3} + \sqrt{3} \cdot 2}{2}$

Step 3.2.2.2

Move $2$ to the left of $\sqrt{3}$ .

$\frac{\sqrt{3} + 2 \sqrt{3}}{2}$

Step 3.2.3

Add $\sqrt{3}$ and $2 \sqrt{3}$ .

$\frac{3 \sqrt{3}}{2}$

Step 4

Replace the summations with the values found.

$\frac{55 \sqrt{3}}{2} - \frac{3 \sqrt{3}}{2}$

Step 5

Simplify.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{55 \sqrt{3} - 3 \sqrt{3}}{2}$

Step 5.2

Subtract $3 \sqrt{3}$ from $55 \sqrt{3}$ .

$\frac{52 \sqrt{3}}{2}$

Step 5.3

Cancel the common factor of $52$ and $2$ .

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Step 5.3.1

Factor $2$ out of $52 \sqrt{3}$ .

$\frac{2 (26 \sqrt{3})}{2}$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $2$ out of $2$ .

$\frac{2 (26 \sqrt{3})}{2 (1)}$

Step 5.3.2.2

Cancel the common factor.

$\frac{2 (26 \sqrt{3})}{2 \cdot 1}$

Step 5.3.2.3

Rewrite the expression.

$\frac{26 \sqrt{3}}{1}$

Step 5.3.2.4

Divide $26 \sqrt{3}$ by $1$ .

$26 \sqrt{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$26 \sqrt{3}$

Decimal Form:

$45.03332099 \dots$