Calculus Examples

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

Step 1

Simplify the summation.

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Step 1.1

Use the Binomial Theorem.

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

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $3 i$ .

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

Step 1.2.2

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

$27 i^{3} + 3 {(3 i)}^{2} \cdot - 2 + 3 (3 i) {(- 2)}^{2} + {(- 2)}^{3}$

Step 1.2.3

Apply the product rule to $3 i$ .

$27 i^{3} + 3 (3^{2} i^{2}) \cdot - 2 + 3 (3 i) {(- 2)}^{2} + {(- 2)}^{3}$

Step 1.2.4

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 1.2.4.1

Move $3^{2}$ .

$27 i^{3} + 3^{2} \cdot 3 i^{2} \cdot - 2 + 3 (3 i) {(- 2)}^{2} + {(- 2)}^{3}$

Step 1.2.4.2

Multiply $3^{2}$ by $3$ .

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Step 1.2.4.2.1

Raise $3$ to the power of $1$ .

$27 i^{3} + 3^{2} \cdot 3^{1} i^{2} \cdot - 2 + 3 (3 i) {(- 2)}^{2} + {(- 2)}^{3}$

Step 1.2.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$27 i^{3} + 3^{2 + 1} i^{2} \cdot - 2 + 3 (3 i) {(- 2)}^{2} + {(- 2)}^{3}$

Step 1.2.4.3

Add $2$ and $1$ .

$27 i^{3} + 3^{3} i^{2} \cdot - 2 + 3 (3 i) {(- 2)}^{2} + {(- 2)}^{3}$

Step 1.2.5

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

$27 i^{3} + 27 i^{2} \cdot - 2 + 3 (3 i) {(- 2)}^{2} + {(- 2)}^{3}$

Step 1.2.6

Multiply $- 2$ by $27$ .

$27 i^{3} - 54 i^{2} + 3 (3 i) {(- 2)}^{2} + {(- 2)}^{3}$

Step 1.2.7

Multiply $3$ by $3$ .

$27 i^{3} - 54 i^{2} + 9 i {(- 2)}^{2} + {(- 2)}^{3}$

Step 1.2.8

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

$27 i^{3} - 54 i^{2} + 9 i \cdot 4 + {(- 2)}^{3}$

Step 1.2.9

Multiply $4$ by $9$ .

$27 i^{3} - 54 i^{2} + 36 i + {(- 2)}^{3}$

Step 1.2.10

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

$27 i^{3} - 54 i^{2} + 36 i - 8$

Step 1.3

Rewrite the summation.

$\sum_{i = 1}^{7} 27 i^{3} - 54 i^{2} + 36 i - 8$

Step 2

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

$\sum_{i = 1}^{7} 27 i^{3} - 54 i^{2} + 36 i - 8 = 27 \sum_{i = 1}^{7} i^{3} - 54 \sum_{i = 1}^{7} i^{2} + 36 \sum_{i = 1}^{7} i + \sum_{i = 1}^{7} - 8$

Step 3

Evaluate $27 \sum_{i = 1}^{7} i^{3}$ .

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Step 3.1

The formula for the summation of a polynomial with degree $3$ is:

$\sum_{k = 1}^{n} i^{3} = \frac{n^{2} {(n + 1)}^{2}}{4}$

Step 3.2

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

$(27) (\frac{7^{2} {(7 + 1)}^{2}}{4})$

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 $1$ .

$27 \frac{7^{2} \cdot 8^{2}}{4}$

Step 3.3.1.2

Raise $7$ to the power of $2$ .

$27 \frac{49 \cdot 8^{2}}{4}$

Step 3.3.1.3

Raise $8$ to the power of $2$ .

$27 \frac{49 \cdot 64}{4}$

Step 3.3.2

Simplify the expression.

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Step 3.3.2.1

Multiply $49$ by $64$ .

$27 (\frac{3136}{4})$

Step 3.3.2.2

Divide $3136$ by $4$ .

$27 \cdot 784$

Step 3.3.2.3

Multiply $27$ by $784$ .

$21168$

Step 4

Evaluate $54 \sum_{i = 1}^{7} i^{2}$ .

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Step 4.1

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 4.2

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

$(- 54) (\frac{7 (7 + 1) (2 \cdot 7 + 1)}{6})$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the numerator.

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Step 4.3.1.1

Add $7$ and $1$ .

$- 54 \frac{7 \cdot 8 (2 \cdot 7 + 1)}{6}$

Step 4.3.1.2

Combine exponents.

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Step 4.3.1.2.1

Multiply $7$ by $8$ .

$- 54 \frac{56 (2 \cdot 7 + 1)}{6}$

Step 4.3.1.2.2

Multiply $2$ by $7$ .

$- 54 \frac{56 (14 + 1)}{6}$

Step 4.3.1.3

Add $14$ and $1$ .

$- 54 \frac{56 \cdot 15}{6}$

Step 4.3.2

Reduce the expression by cancelling the common factors.

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Step 4.3.2.1

Multiply $56$ by $15$ .

$- 54 (\frac{840}{6})$

Step 4.3.2.2

Cancel the common factor of $6$ .

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Step 4.3.2.2.1

Factor $6$ out of $- 54$ .

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

Step 4.3.2.2.2

Cancel the common factor.

$6 \cdot - 9 \frac{840}{6}$

Step 4.3.2.2.3

Rewrite the expression.

$- 9 \cdot 840$

Step 4.3.2.3

Multiply $- 9$ by $840$ .

$- 7560$

Step 5

Evaluate $36 \sum_{i = 1}^{7} i$ .

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Step 5.1

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

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

Step 5.2

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

$(36) (\frac{7 (7 + 1)}{2})$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the expression.

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Step 5.3.1.1

Add $7$ and $1$ .

$36 \frac{7 \cdot 8}{2}$

Step 5.3.1.2

Multiply $7$ by $8$ .

$36 (\frac{56}{2})$

Step 5.3.2

Cancel the common factor of $2$ .

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Step 5.3.2.1

Factor $2$ out of $36$ .

$2 (18) \frac{56}{2}$

Step 5.3.2.2

Cancel the common factor.

$2 \cdot 18 \frac{56}{2}$

Step 5.3.2.3

Rewrite the expression.

$18 \cdot 56$

Step 5.3.3

Multiply $18$ by $56$ .

$1008$

Step 6

Evaluate $\sum_{i = 1}^{7} - 8$ .

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Step 6.1

The formula for the summation of a constant is:

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

Step 6.2

Substitute the values into the formula.

$(- 8) (7)$

Step 6.3

Multiply $- 8$ by $7$ .

$- 56$

Step 7

Add the results of the summations.

$21168 - 7560 + 1008 - 56$

Step 8

Simplify.

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Step 8.1

Subtract $7560$ from $21168$ .

$13608 + 1008 - 56$

Step 8.2

Add $13608$ and $1008$ .

$14616 - 56$

Step 8.3

Subtract $56$ from $14616$ .

$14560$