Calculus Examples

$7 \sum_{t = 4}^{9} t^{2} - 7 t$

Step 1

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

$7 \sum_{t = 4}^{9} t^{2} - 7 t = 7 (\sum_{t = 1}^{9} t^{2} - 7 t - \sum_{t = 1}^{3} t^{2} - 7 t)$

Step 2

Evaluate $\sum_{t = 1}^{9} t^{2} - 7 t$ .

Tap for more steps...

Step 2.1

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

$\sum_{t = 1}^{9} t^{2} - 7 t = \sum_{t = 1}^{9} t^{2} - 7 \sum_{t = 1}^{9} t$

Step 2.2

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

Tap for more steps...

Step 2.2.1

The formula for the summation of a polynomial with degree $2$ is:

$\sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

Step 2.2.2

Substitute the values into the formula.

$\frac{9 (9 + 1) (2 \cdot 9 + 1)}{6}$

Step 2.2.3

Simplify.

Tap for more steps...

Step 2.2.3.1

Cancel the common factor of $9$ and $6$ .

Tap for more steps...

Step 2.2.3.1.1

Factor $3$ out of $9 (9 + 1) (2 \cdot 9 + 1)$ .

$\frac{3 (3 (9 + 1) (2 \cdot 9 + 1))}{6}$

Step 2.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.2.3.1.2.1

Factor $3$ out of $6$ .

$\frac{3 (3 (9 + 1) (2 \cdot 9 + 1))}{3 (2)}$

Step 2.2.3.1.2.2

Cancel the common factor.

$\frac{3 (3 (9 + 1) (2 \cdot 9 + 1))}{3 \cdot 2}$

Step 2.2.3.1.2.3

Rewrite the expression.

$\frac{3 (9 + 1) (2 \cdot 9 + 1)}{2}$

Step 2.2.3.2

Simplify the numerator.

Tap for more steps...

Step 2.2.3.2.1

Multiply $2$ by $9$ .

$\frac{3 (9 + 1) (18 + 1)}{2}$

Step 2.2.3.2.2

Add $9$ and $1$ .

$\frac{3 \cdot 10 (18 + 1)}{2}$

Step 2.2.3.2.3

Multiply $3$ by $10$ .

$\frac{30 (18 + 1)}{2}$

Step 2.2.3.2.4

Add $18$ and $1$ .

$\frac{30 \cdot 19}{2}$

Step 2.2.3.3

Simplify the expression.

Tap for more steps...

Step 2.2.3.3.1

Multiply $30$ by $19$ .

$\frac{570}{2}$

Step 2.2.3.3.2

Divide $570$ by $2$ .

$285$

Step 2.3

Evaluate $7 \sum_{t = 1}^{9} t$ .

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.

$(- 7) (\frac{9 (9 + 1)}{2})$

Step 2.3.3

Simplify.

Tap for more steps...

Step 2.3.3.1

Add $9$ and $1$ .

$- 7 \frac{9 \cdot 10}{2}$

Step 2.3.3.2

Multiply $9$ by $10$ .

$- 7 (\frac{90}{2})$

Step 2.3.3.3

Divide $90$ by $2$ .

$- 7 \cdot 45$

Step 2.3.3.4

Multiply $- 7$ by $45$ .

$- 315$

Step 2.4

Add the results of the summations.

$285 - 315$

Step 2.5

Subtract $315$ from $285$ .

$- 30$

Step 3

Evaluate $\sum_{t = 1}^{3} t^{2} - 7 t$ .

Tap for more steps...

Step 3.1

Expand the series for each value of $t$ .

$1^{2} - 7 \cdot 1 + 2^{2} - 7 \cdot 2 + 3^{2} - 7 \cdot 3$

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

Raise $1$ to the power of $2$ .

$1 - 7 \cdot 1 + 2^{2} - 7 \cdot 2 + 3^{2} - 7 \cdot 3$

Step 3.2.2

Multiply $- 7$ by $1$ .

$1 - 7 + 2^{2} - 7 \cdot 2 + 3^{2} - 7 \cdot 3$

Step 3.2.3

Subtract $7$ from $1$ .

$- 6 + 2^{2} - 7 \cdot 2 + 3^{2} - 7 \cdot 3$

Step 3.2.4

Raise $2$ to the power of $2$ .

$- 6 + 4 - 7 \cdot 2 + 3^{2} - 7 \cdot 3$

Step 3.2.5

Multiply $- 7$ by $2$ .

$- 6 + 4 - 14 + 3^{2} - 7 \cdot 3$

Step 3.2.6

Subtract $14$ from $4$ .

$- 6 - 10 + 3^{2} - 7 \cdot 3$

Step 3.2.7

Subtract $10$ from $- 6$ .

$- 16 + 3^{2} - 7 \cdot 3$

Step 3.2.8

Raise $3$ to the power of $2$ .

$- 16 + 9 - 7 \cdot 3$

Step 3.2.9

Multiply $- 7$ by $3$ .

$- 16 + 9 - 21$

Step 3.2.10

Subtract $21$ from $9$ .

$- 16 - 12$

Step 3.2.11

Subtract $12$ from $- 16$ .

$- 28$

Step 4

Replace the summations with the values found.

$7 (- 30 + 28)$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Add $- 30$ and $28$ .

$7 \cdot - 2$

Step 5.2

Multiply $7$ by $- 2$ .

$- 14$