Calculus Examples

$\sum_{i = 1}^{50} (5 i - 2) (i + 3)$

Step 1

Simplify the summation.

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Step 1.1

Expand $(5 i - 2) (i + 3)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$5 i (i + 3) - 2 (i + 3)$

Step 1.1.2

Apply the distributive property.

$5 i \cdot i + 5 i \cdot 3 - 2 (i + 3)$

Step 1.1.3

Apply the distributive property.

$5 i \cdot i + 5 i \cdot 3 - 2 i - 2 \cdot 3$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $i$ by $i$ by adding the exponents.

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Step 1.2.1.1.1

Move $i$ .

$5 (i \cdot i) + 5 i \cdot 3 - 2 i - 2 \cdot 3$

Step 1.2.1.1.2

Multiply $i$ by $i$ .

$5 i^{2} + 5 i \cdot 3 - 2 i - 2 \cdot 3$

Step 1.2.1.2

Multiply $3$ by $5$ .

$5 i^{2} + 15 i - 2 i - 2 \cdot 3$

Step 1.2.1.3

Multiply $- 2$ by $3$ .

$5 i^{2} + 15 i - 2 i - 6$

Step 1.2.2

Subtract $2 i$ from $15 i$ .

$5 i^{2} + 13 i - 6$

Step 1.3

Rewrite the summation.

$\sum_{i = 1}^{50} 5 i^{2} + 13 i - 6$

Step 2

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

$\sum_{i = 1}^{50} 5 i^{2} + 13 i - 6 = 5 \sum_{i = 1}^{50} i^{2} + 13 \sum_{i = 1}^{50} i + \sum_{i = 1}^{50} - 6$

Step 3

Evaluate $5 \sum_{i = 1}^{50} i^{2}$ .

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Step 3.1

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

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

Step 3.2

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

$(5) (\frac{50 (50 + 1) (2 \cdot 50 + 1)}{6})$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Add $50$ and $1$ .

$5 \frac{50 \cdot 51 (2 \cdot 50 + 1)}{6}$

Step 3.3.1.2

Combine exponents.

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Step 3.3.1.2.1

Multiply $50$ by $51$ .

$5 \frac{2550 (2 \cdot 50 + 1)}{6}$

Step 3.3.1.2.2

Multiply $2$ by $50$ .

$5 \frac{2550 (100 + 1)}{6}$

Step 3.3.1.3

Add $100$ and $1$ .

$5 \frac{2550 \cdot 101}{6}$

Step 3.3.2

Simplify the expression.

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Step 3.3.2.1

Multiply $2550$ by $101$ .

$5 (\frac{257550}{6})$

Step 3.3.2.2

Divide $257550$ by $6$ .

$5 \cdot 42925$

Step 3.3.2.3

Multiply $5$ by $42925$ .

$214625$

Step 4

Evaluate $13 \sum_{i = 1}^{50} i$ .

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Step 4.1

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

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

Step 4.2

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

$(13) (\frac{50 (50 + 1)}{2})$

Step 4.3

Simplify.

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Step 4.3.1

Add $50$ and $1$ .

$13 \frac{50 \cdot 51}{2}$

Step 4.3.2

Multiply $50$ by $51$ .

$13 (\frac{2550}{2})$

Step 4.3.3

Divide $2550$ by $2$ .

$13 \cdot 1275$

Step 4.3.4

Multiply $13$ by $1275$ .

$16575$

Step 5

Evaluate $\sum_{i = 1}^{50} - 6$ .

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Step 5.1

The formula for the summation of a constant is:

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

Step 5.2

Substitute the values into the formula.

$(- 6) (50)$

Step 5.3

Multiply $- 6$ by $50$ .

$- 300$

Step 6

Add the results of the summations.

$214625 + 16575 - 300$

Step 7

Simplify.

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Step 7.1

Add $214625$ and $16575$ .

$231200 - 300$

Step 7.2

Subtract $300$ from $231200$ .

$230900$