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

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

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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

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

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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.

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Step 2.2.3.1

Cancel the common factor of

$9$ and

$6$ .

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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.

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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.

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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.

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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

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

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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.

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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

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

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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.

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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.

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Step 5.1

Add

$- 30$ and

$28$ .

$7 \cdot - 2$

Step 5.2

Multiply

$7$ by

$- 2$ .

$- 14$