Calculus Examples

$\sum_{i = 0}^{3} {(3 i + 2)}^{2}$

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

Add $6 i$ and $6 i$ .

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

Step 1.4

Rewrite the summation.

$\sum_{i = 0}^{3} 9 i^{2} + 12 i + 4$

Step 2

Expand the series for each value of $i$ .

$9 \cdot 0^{2} + 12 \cdot 0 + 4 + 9 \cdot 1^{2} + 12 \cdot 1 + 4 + 9 \cdot 2^{2} + 12 \cdot 2 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3

Simplify.

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Step 3.1

Raise $0$ to the power of $2$ .

$9 \cdot 0 + 12 \cdot 0 + 4 + 9 \cdot 1^{2} + 12 \cdot 1 + 4 + 9 \cdot 2^{2} + 12 \cdot 2 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.2

Multiply $9$ by $0$ .

$0 + 12 \cdot 0 + 4 + 9 \cdot 1^{2} + 12 \cdot 1 + 4 + 9 \cdot 2^{2} + 12 \cdot 2 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.3

Multiply $12$ by $0$ .

$0 + 0 + 4 + 9 \cdot 1^{2} + 12 \cdot 1 + 4 + 9 \cdot 2^{2} + 12 \cdot 2 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.4

Add $0$ and $0$ .

$0 + 4 + 9 \cdot 1^{2} + 12 \cdot 1 + 4 + 9 \cdot 2^{2} + 12 \cdot 2 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.5

Add $0$ and $4$ .

$4 + 9 \cdot 1^{2} + 12 \cdot 1 + 4 + 9 \cdot 2^{2} + 12 \cdot 2 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.6

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

$4 + 9 \cdot 1 + 12 \cdot 1 + 4 + 9 \cdot 2^{2} + 12 \cdot 2 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.7

Multiply $9$ by $1$ .

$4 + 9 + 12 \cdot 1 + 4 + 9 \cdot 2^{2} + 12 \cdot 2 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.8

Multiply $12$ by $1$ .

$4 + 9 + 12 + 4 + 9 \cdot 2^{2} + 12 \cdot 2 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.9

Add $9$ and $12$ .

$4 + 21 + 4 + 9 \cdot 2^{2} + 12 \cdot 2 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.10

Add $21$ and $4$ .

$4 + 25 + 9 \cdot 2^{2} + 12 \cdot 2 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.11

Add $4$ and $25$ .

$29 + 9 \cdot 2^{2} + 12 \cdot 2 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.12

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

$29 + 9 \cdot 4 + 12 \cdot 2 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.13

Multiply $9$ by $4$ .

$29 + 36 + 12 \cdot 2 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.14

Multiply $12$ by $2$ .

$29 + 36 + 24 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.15

Add $36$ and $24$ .

$29 + 60 + 4 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.16

Add $60$ and $4$ .

$29 + 64 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.17

Add $29$ and $64$ .

$93 + 9 \cdot 3^{2} + 12 \cdot 3 + 4$

Step 3.18

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

$93 + 9 \cdot 9 + 12 \cdot 3 + 4$

Step 3.19

Multiply $9$ by $9$ .

$93 + 81 + 12 \cdot 3 + 4$

Step 3.20

Multiply $12$ by $3$ .

$93 + 81 + 36 + 4$

Step 3.21

Add $81$ and $36$ .

$93 + 117 + 4$

Step 3.22

Add $117$ and $4$ .

$93 + 121$

Step 3.23

Add $93$ and $121$ .

$214$