Calculus Examples

$\sum_{k = 1}^{25} \frac{11}{3} - \frac{5}{3} k$

Step 1

Simplify the summation.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $k$ and $\frac{5}{3}$ .

$\frac{11}{3} - \frac{k \cdot 5}{3}$

Step 1.1.2

Move $5$ to the left of $k$ .

$\frac{11}{3} - \frac{5 k}{3}$

Step 1.2

Rewrite the summation.

$\sum_{k = 1}^{25} \frac{11}{3} - \frac{5 k}{3}$

Step 2

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

$\sum_{k = 1}^{25} \frac{11}{3} - \frac{5 k}{3} = \sum_{k = 1}^{25} \frac{11}{3} + \sum_{k = 1}^{25} - \frac{5 k}{3}$

Step 3

Evaluate $\sum_{k = 1}^{25} \frac{11}{3}$ .

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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.

$(\frac{11}{3}) (25)$

Step 3.3

Multiply $(\frac{11}{3}) (25)$ .

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Step 3.3.1

Combine $\frac{11}{3}$ and $25$ .

$\frac{11 \cdot 25}{3}$

Step 3.3.2

Multiply $11$ by $25$ .

$\frac{275}{3}$

Step 4

Evaluate $\sum_{k = 1}^{25} - \frac{5 k}{3}$ .

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Step 4.1

Factor $- \frac{5}{3}$ out of the summation.

$(- \frac{5}{3}) \sum_{k = 1}^{25} k$

Step 4.2

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

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

Step 4.3

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

$(- \frac{5}{3}) (\frac{25 (25 + 1)}{2})$

Step 4.4

Simplify.

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Step 4.4.1

Add $25$ and $1$ .

$- \frac{5}{3} \cdot \frac{25 \cdot 26}{2}$

Step 4.4.2

Multiply $25$ by $26$ .

$- \frac{5}{3} \cdot \frac{650}{2}$

Step 4.4.3

Divide $650$ by $2$ .

$- \frac{5}{3} \cdot 325$

Step 4.4.4

Multiply $- \frac{5}{3} \cdot 325$ .

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Step 4.4.4.1

Multiply $325$ by $- 1$ .

$- 325 (\frac{5}{3})$

Step 4.4.4.2

Combine $- 325$ and $\frac{5}{3}$ .

$\frac{- 325 \cdot 5}{3}$

Step 4.4.4.3

Multiply $- 325$ by $5$ .

$\frac{- 1625}{3}$

Step 4.4.5

Move the negative in front of the fraction.

$- \frac{1625}{3}$

Step 5

Add the results of the summations.

$\frac{275}{3} - \frac{1625}{3}$

Step 6

Simplify.

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Step 6.1

Combine the numerators over the common denominator.

$\frac{275 - 1625}{3}$

Step 6.2

Simplify the expression.

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Step 6.2.1

Subtract $1625$ from $275$ .

$\frac{- 1350}{3}$

Step 6.2.2

Divide $- 1350$ by $3$ .

$- 450$