Calculus Examples

$\sum_{k = 1}^{4} {(k + 7)}^{2}$

Step 1

Simplify the summation.

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Step 1.1

Rewrite ${(k + 7)}^{2}$ as $(k + 7) (k + 7)$ .

$(k + 7) (k + 7)$

Step 1.2

Expand $(k + 7) (k + 7)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$k (k + 7) + 7 (k + 7)$

Step 1.2.2

Apply the distributive property.

$k \cdot k + k \cdot 7 + 7 (k + 7)$

Step 1.2.3

Apply the distributive property.

$k \cdot k + k \cdot 7 + 7 k + 7 \cdot 7$

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

Multiply $k$ by $k$ .

$k^{2} + k \cdot 7 + 7 k + 7 \cdot 7$

Step 1.3.1.2

Move $7$ to the left of $k$ .

$k^{2} + 7 \cdot k + 7 k + 7 \cdot 7$

Step 1.3.1.3

Multiply $7$ by $7$ .

$k^{2} + 7 k + 7 k + 49$

Step 1.3.2

Add $7 k$ and $7 k$ .

$k^{2} + 14 k + 49$

Step 1.4

Rewrite the summation.

$\sum_{k = 1}^{4} k^{2} + 14 k + 49$

Step 2

Expand the series for each value of $k$ .

$1^{2} + 14 \cdot 1 + 49 + 2^{2} + 14 \cdot 2 + 49 + 3^{2} + 14 \cdot 3 + 49 + 4^{2} + 14 \cdot 4 + 49$

Step 3

Simplify.

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Step 3.1

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

$1 + 14 \cdot 1 + 49 + 2^{2} + 14 \cdot 2 + 49 + 3^{2} + 14 \cdot 3 + 49 + 4^{2} + 14 \cdot 4 + 49$

Step 3.2

Multiply $14$ by $1$ .

$1 + 14 + 49 + 2^{2} + 14 \cdot 2 + 49 + 3^{2} + 14 \cdot 3 + 49 + 4^{2} + 14 \cdot 4 + 49$

Step 3.3

Add $1$ and $14$ .

$15 + 49 + 2^{2} + 14 \cdot 2 + 49 + 3^{2} + 14 \cdot 3 + 49 + 4^{2} + 14 \cdot 4 + 49$

Step 3.4

Add $15$ and $49$ .

$64 + 2^{2} + 14 \cdot 2 + 49 + 3^{2} + 14 \cdot 3 + 49 + 4^{2} + 14 \cdot 4 + 49$

Step 3.5

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

$64 + 4 + 14 \cdot 2 + 49 + 3^{2} + 14 \cdot 3 + 49 + 4^{2} + 14 \cdot 4 + 49$

Step 3.6

Multiply $14$ by $2$ .

$64 + 4 + 28 + 49 + 3^{2} + 14 \cdot 3 + 49 + 4^{2} + 14 \cdot 4 + 49$

Step 3.7

Add $4$ and $28$ .

$64 + 32 + 49 + 3^{2} + 14 \cdot 3 + 49 + 4^{2} + 14 \cdot 4 + 49$

Step 3.8

Add $32$ and $49$ .

$64 + 81 + 3^{2} + 14 \cdot 3 + 49 + 4^{2} + 14 \cdot 4 + 49$

Step 3.9

Add $64$ and $81$ .

$145 + 3^{2} + 14 \cdot 3 + 49 + 4^{2} + 14 \cdot 4 + 49$

Step 3.10

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

$145 + 9 + 14 \cdot 3 + 49 + 4^{2} + 14 \cdot 4 + 49$

Step 3.11

Multiply $14$ by $3$ .

$145 + 9 + 42 + 49 + 4^{2} + 14 \cdot 4 + 49$

Step 3.12

Add $9$ and $42$ .

$145 + 51 + 49 + 4^{2} + 14 \cdot 4 + 49$

Step 3.13

Add $51$ and $49$ .

$145 + 100 + 4^{2} + 14 \cdot 4 + 49$

Step 3.14

Add $145$ and $100$ .

$245 + 4^{2} + 14 \cdot 4 + 49$

Step 3.15

Raise $4$ to the power of $2$ .

$245 + 16 + 14 \cdot 4 + 49$

Step 3.16

Multiply $14$ by $4$ .

$245 + 16 + 56 + 49$

Step 3.17

Add $16$ and $56$ .

$245 + 72 + 49$

Step 3.18

Add $72$ and $49$ .

$245 + 121$

Step 3.19

Add $245$ and $121$ .

$366$