Calculus Examples

$\sum_{i = 1}^{7} {(3 i - 2)}^{2}$

Step 1

Simplify the summation.

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Step 1.1

Rewrite ${(3 i - 2)}^{2}$ as $(3 i - 2) (3 i - 2)$ .

$(3 i - 2) (3 i - 2)$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$3 i (3 i - 2) - 2 (3 i - 2)$

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $i$ .

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

Step 1.3.1.2.2

Multiply $i$ by $i$ .

$3 \cdot 3 i^{2} + 3 i \cdot - 2 - 2 (3 i) - 2 \cdot - 2$

Step 1.3.1.3

Multiply $3$ by $3$ .

$9 i^{2} + 3 i \cdot - 2 - 2 (3 i) - 2 \cdot - 2$

Step 1.3.1.4

Multiply $- 2$ by $3$ .

$9 i^{2} - 6 i - 2 (3 i) - 2 \cdot - 2$

Step 1.3.1.5

Multiply $3$ by $- 2$ .

$9 i^{2} - 6 i - 6 i - 2 \cdot - 2$

Step 1.3.1.6

Multiply $- 2$ by $- 2$ .

$9 i^{2} - 6 i - 6 i + 4$

Step 1.3.2

Subtract $6 i$ from $- 6 i$ .

$9 i^{2} - 12 i + 4$

Step 1.4

Rewrite the summation.

$\sum_{i = 1}^{7} 9 i^{2} - 12 i + 4$

Step 2

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

$\sum_{i = 1}^{7} 9 i^{2} - 12 i + 4 = 9 \sum_{i = 1}^{7} i^{2} - 12 \sum_{i = 1}^{7} i + \sum_{i = 1}^{7} 4$

Step 3

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

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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.

$(9) (\frac{7 (7 + 1) (2 \cdot 7 + 1)}{6})$

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 $7$ and $1$ .

$9 \frac{7 \cdot 8 (2 \cdot 7 + 1)}{6}$

Step 3.3.1.2

Combine exponents.

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Step 3.3.1.2.1

Multiply $7$ by $8$ .

$9 \frac{56 (2 \cdot 7 + 1)}{6}$

Step 3.3.1.2.2

Multiply $2$ by $7$ .

$9 \frac{56 (14 + 1)}{6}$

Step 3.3.1.3

Add $14$ and $1$ .

$9 \frac{56 \cdot 15}{6}$

Step 3.3.2

Simplify terms.

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Step 3.3.2.1

Multiply $56$ by $15$ .

$9 (\frac{840}{6})$

Step 3.3.2.2

Cancel the common factor of $3$ .

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Step 3.3.2.2.1

Factor $3$ out of $9$ .

$3 (3) \frac{840}{6}$

Step 3.3.2.2.2

Factor $3$ out of $6$ .

$3 \cdot 3 \frac{840}{3 \cdot 2}$

Step 3.3.2.2.3

Cancel the common factor.

$3 \cdot 3 \frac{840}{3 \cdot 2}$

Step 3.3.2.2.4

Rewrite the expression.

$3 (\frac{840}{2})$

Step 3.3.2.3

Combine $3$ and $\frac{840}{2}$ .

$\frac{3 \cdot 840}{2}$

Step 3.3.2.4

Simplify the expression.

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Step 3.3.2.4.1

Multiply $3$ by $840$ .

$\frac{2520}{2}$

Step 3.3.2.4.2

Divide $2520$ by $2$ .

$1260$

Step 4

Evaluate $12 \sum_{i = 1}^{7} i$ .

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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.

$(- 12) (\frac{7 (7 + 1)}{2})$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the expression.

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Step 4.3.1.1

Add $7$ and $1$ .

$- 12 \frac{7 \cdot 8}{2}$

Step 4.3.1.2

Multiply $7$ by $8$ .

$- 12 (\frac{56}{2})$

Step 4.3.2

Cancel the common factor of $2$ .

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Step 4.3.2.1

Factor $2$ out of $- 12$ .

$2 (- 6) \frac{56}{2}$

Step 4.3.2.2

Cancel the common factor.

$2 \cdot - 6 \frac{56}{2}$

Step 4.3.2.3

Rewrite the expression.

$- 6 \cdot 56$

Step 4.3.3

Multiply $- 6$ by $56$ .

$- 336$

Step 5

Evaluate $\sum_{i = 1}^{7} 4$ .

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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.

$(4) (7)$

Step 5.3

Multiply $4$ by $7$ .

$28$

Step 6

Add the results of the summations.

$1260 - 336 + 28$

Step 7

Simplify.

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Step 7.1

Subtract $336$ from $1260$ .

$924 + 28$

Step 7.2

Add $924$ and $28$ .

$952$