Precalculus Examples

$\sum_{k = 4}^{21} 59 - 7 k$

Step 1

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

$\sum_{k = 4}^{21} 59 - 7 k = \sum_{k = 1}^{21} 59 - 7 k - \sum_{k = 1}^{3} 59 - 7 k$

Step 2

Evaluate $\sum_{k = 1}^{21} 59 - 7 k$ .

Tap for more steps...

Step 2.1

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

$\sum_{k = 1}^{21} 59 - 7 k = \sum_{k = 1}^{21} 59 - 7 \sum_{k = 1}^{21} k$

Step 2.2

Evaluate $\sum_{k = 1}^{21} 59$ .

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.

$(59) (21)$

Step 2.2.3

Multiply $59$ by $21$ .

$1239$

Step 2.3

Evaluate $7 \sum_{k = 1}^{21} k$ .

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

$(- 7) (\frac{21 (21 + 1)}{2})$

Step 2.3.3

Simplify.

Tap for more steps...

Step 2.3.3.1

Add $21$ and $1$ .

$- 7 \frac{21 \cdot 22}{2}$

Step 2.3.3.2

Multiply $21$ by $22$ .

$- 7 (\frac{462}{2})$

Step 2.3.3.3

Divide $462$ by $2$ .

$- 7 \cdot 231$

Step 2.3.3.4

Multiply $- 7$ by $231$ .

$- 1617$

Step 2.4

Add the results of the summations.

$1239 - 1617$

Step 2.5

Subtract $1617$ from $1239$ .

$- 378$

Step 3

Evaluate $\sum_{k = 1}^{3} 59 - 7 k$ .

Tap for more steps...

Step 3.1

Expand the series for each value of $k$ .

$59 - 7 \cdot 1 + 59 - 7 \cdot 2 + 59 - 7 \cdot 3$

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

Multiply $- 7$ by $1$ .

$59 - 7 + 59 - 7 \cdot 2 + 59 - 7 \cdot 3$

Step 3.2.2

Subtract $7$ from $59$ .

$52 + 59 - 7 \cdot 2 + 59 - 7 \cdot 3$

Step 3.2.3

Multiply $- 7$ by $2$ .

$52 + 59 - 14 + 59 - 7 \cdot 3$

Step 3.2.4

Subtract $14$ from $59$ .

$52 + 45 + 59 - 7 \cdot 3$

Step 3.2.5

Add $52$ and $45$ .

$97 + 59 - 7 \cdot 3$

Step 3.2.6

Multiply $- 7$ by $3$ .

$97 + 59 - 21$

Step 3.2.7

Subtract $21$ from $59$ .

$97 + 38$

Step 3.2.8

Add $97$ and $38$ .

$135$

Step 4

Replace the summations with the values found.

$- 378 - 135$

Step 5

Subtract $135$ from $- 378$ .

$- 513$