Calculus Examples

$\sum_{1}^{225} 3 x - 7$

Step 1

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

$\sum_{1}^{225} 3 x - 7 = 3 \sum_{x = 1}^{225} x + \sum_{x = 1}^{225} - 7$

Step 2

Evaluate $3 \sum_{x = 1}^{225} x$ .

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Step 2.1

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

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

Step 2.2

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

$(3) (\frac{225 (225 + 1)}{2})$

Step 2.3

Simplify.

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Step 2.3.1

Add $225$ and $1$ .

$3 \frac{225 \cdot 226}{2}$

Step 2.3.2

Multiply $225$ by $226$ .

$3 (\frac{50850}{2})$

Step 2.3.3

Divide $50850$ by $2$ .

$3 \cdot 25425$

Step 2.3.4

Multiply $3$ by $25425$ .

$76275$

Step 3

Evaluate $\sum_{x = 1}^{225} - 7$ .

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Step 3.1

The formula for the summation of a constant is:

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

Step 3.2

Substitute the values into the formula.

$(- 7) (225)$

Step 3.3

Multiply $- 7$ by $225$ .

$- 1575$

Step 4

Add the results of the summations.

$76275 - 1575$

Step 5

Subtract $1575$ from $76275$ .

$74700$