Precalculus Examples

$\sum_{i = 10}^{25} - 2 - 8 i$

Step 1

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

$\sum_{k = 10}^{25} - 2 - 8 i = \sum_{k = 1}^{25} - 2 - 8 i - \sum_{k = 1}^{9} - 2 - 8 i$

Step 2

Evaluate $\sum_{k = 1}^{25} - 2 - 8 i$ .

Tap for more steps...

Step 2.1

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

$\sum_{k = 1}^{25} - 2 - 8 i = \sum_{k = 1}^{25} - 2 - 8 \sum_{k = 1}^{25} i$

Step 2.2

Evaluate $\sum_{k = 1}^{25} - 2$ .

Tap for more steps...

Step 2.2.1

The formula for the summation of a constant is:

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

Step 2.2.2

Substitute the values into the formula.

$(- 2) (25)$

Step 2.2.3

Multiply $- 2$ by $25$ .

$- 50$

Step 2.3

Evaluate $8 \sum_{k = 1}^{25} i$ .

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

$(- 8) (\frac{25 (25 + 1)}{2})$

Step 2.3.3

Simplify.

Tap for more steps...

Step 2.3.3.1

Simplify the expression.

Tap for more steps...

Step 2.3.3.1.1

Add $25$ and $1$ .

$- 8 \frac{25 \cdot 26}{2}$

Step 2.3.3.1.2

Multiply $25$ by $26$ .

$- 8 (\frac{650}{2})$

Step 2.3.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.3.2.1

Factor $2$ out of $- 8$ .

$2 (- 4) \frac{650}{2}$

Step 2.3.3.2.2

Cancel the common factor.

$2 \cdot - 4 \frac{650}{2}$

Step 2.3.3.2.3

Rewrite the expression.

$- 4 \cdot 650$

Step 2.3.3.3

Multiply $- 4$ by $650$ .

$- 2600$

Step 2.4

Add the results of the summations.

$- 50 - 2600$

Step 2.5

Subtract $2600$ from $- 50$ .

$- 2650$

Step 3

Evaluate $\sum_{k = 1}^{9} - 2 - 8 i$ .

Tap for more steps...

Step 3.1

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

$\sum_{k = 1}^{9} - 2 - 8 i = \sum_{k = 1}^{9} - 2 - 8 \sum_{k = 1}^{9} i$

Step 3.2

Evaluate $\sum_{k = 1}^{9} - 2$ .

Tap for more steps...

Step 3.2.1

The formula for the summation of a constant is:

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

Step 3.2.2

Substitute the values into the formula.

$(- 2) (9)$

Step 3.2.3

Multiply $- 2$ by $9$ .

$- 18$

Step 3.3

Evaluate $8 \sum_{k = 1}^{9} i$ .

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

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

$(- 8) (\frac{9 (9 + 1)}{2})$

Step 3.3.3

Simplify.

Tap for more steps...

Step 3.3.3.1

Simplify the expression.

Tap for more steps...

Step 3.3.3.1.1

Add $9$ and $1$ .

$- 8 \frac{9 \cdot 10}{2}$

Step 3.3.3.1.2

Multiply $9$ by $10$ .

$- 8 (\frac{90}{2})$

Step 3.3.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.3.2.1

Factor $2$ out of $- 8$ .

$2 (- 4) \frac{90}{2}$

Step 3.3.3.2.2

Cancel the common factor.

$2 \cdot - 4 \frac{90}{2}$

Step 3.3.3.2.3

Rewrite the expression.

$- 4 \cdot 90$

Step 3.3.3.3

Multiply $- 4$ by $90$ .

$- 360$

Step 3.4

Add the results of the summations.

$- 18 - 360$

Step 3.5

Subtract $360$ from $- 18$ .

$- 378$

Step 4

Replace the summations with the values found.

$- 2650 + 378$

Step 5

Add $- 2650$ and $378$ .

$- 2272$