Calculus Examples

$\sum_{i = 1}^{25} \frac{i + 1}{625}$

Step 1

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

$\sum_{k = 1}^{25} \frac{i}{625} + \frac{1}{625} = \sum_{k = 1}^{25} \frac{i}{625} + \sum_{k = 1}^{25} \frac{1}{625}$

Step 2

Evaluate $\sum_{k = 1}^{25} \frac{i}{625}$ .

Tap for more steps...

Step 2.1

Factor $\frac{1}{625}$ out of the summation.

$\frac{1}{625} \sum_{k = 1}^{25} i$

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.

$(\frac{1}{625}) (\frac{25 (25 + 1)}{2})$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Simplify the expression.

Tap for more steps...

Step 2.4.1.1

Add $25$ and $1$ .

$\frac{1}{625} \cdot \frac{25 \cdot 26}{2}$

Step 2.4.1.2

Multiply $25$ by $26$ .

$\frac{1}{625} \cdot \frac{650}{2}$

Step 2.4.2

Cancel the common factor of $25$ .

Tap for more steps...

Step 2.4.2.1

Factor $25$ out of $625$ .

$\frac{1}{25 (25)} \cdot \frac{650}{2}$

Step 2.4.2.2

Factor $25$ out of $650$ .

$\frac{1}{25 \cdot 25} \cdot \frac{25 \cdot 26}{2}$

Step 2.4.2.3

Cancel the common factor.

$\frac{1}{25 \cdot 25} \cdot \frac{25 \cdot 26}{2}$

Step 2.4.2.4

Rewrite the expression.

$\frac{1}{25} \cdot \frac{26}{2}$

Step 2.4.3

Multiply $\frac{1}{25}$ by $\frac{26}{2}$ .

$\frac{26}{25 \cdot 2}$

Step 2.4.4

Multiply $25$ by $2$ .

$\frac{26}{50}$

Step 2.4.5

Cancel the common factor of $26$ and $50$ .

Tap for more steps...

Step 2.4.5.1

Factor $2$ out of $26$ .

$\frac{2 (13)}{50}$

Step 2.4.5.2

Cancel the common factors.

Tap for more steps...

Step 2.4.5.2.1

Factor $2$ out of $50$ .

$\frac{2 \cdot 13}{2 \cdot 25}$

Step 2.4.5.2.2

Cancel the common factor.

$\frac{2 \cdot 13}{2 \cdot 25}$

Step 2.4.5.2.3

Rewrite the expression.

$\frac{13}{25}$

Step 3

Evaluate $\sum_{k = 1}^{25} \frac{1}{625}$ .

Tap for more steps...

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.

$(\frac{1}{625}) (25)$

Step 3.3

Cancel the common factor of $25$ .

Tap for more steps...

Step 3.3.1

Factor $25$ out of $625$ .

$\frac{1}{25 (25)} \cdot 25$

Step 3.3.2

Cancel the common factor.

$\frac{1}{25 \cdot 25} \cdot 25$

Step 3.3.3

Rewrite the expression.

$\frac{1}{25}$

Step 4

Add the results of the summations.

$\frac{13}{25} + \frac{1}{25}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Combine the numerators over the common denominator.

$\frac{13 + 1}{25}$

Step 5.2

Add $13$ and $1$ .

$\frac{14}{25}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{14}{25}$

Decimal Form:

$0.56$