Calculus Examples

$\sum_{k = 1}^{8} k (6 k + 7)$

Step 1

Simplify the summation.

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Step 1.1

Apply the distributive property.

$k (6 k) + k \cdot 7$

Step 1.2

Rewrite using the commutative property of multiplication.

$6 k \cdot k + k \cdot 7$

Step 1.3

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

$6 k \cdot k + 7 \cdot k$

Step 1.4

Multiply $k$ by $k$ by adding the exponents.

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Step 1.4.1

Move $k$ .

$6 (k \cdot k) + 7 \cdot k$

Step 1.4.2

Multiply $k$ by $k$ .

$6 k^{2} + 7 \cdot k$

$6 k^{2} + 7 k$

Step 1.5

Rewrite the summation.

$\sum_{k = 1}^{8} 6 k^{2} + 7 k$

Step 2

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

$\sum_{k = 1}^{8} 6 k^{2} + 7 k = 6 \sum_{k = 1}^{8} k^{2} + 7 \sum_{k = 1}^{8} k$

Step 3

Evaluate $6 \sum_{k = 1}^{8} k^{2}$ .

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Step 3.1

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

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

Step 3.2

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

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

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 $8$ and $1$ .

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

Step 3.3.1.2

Combine exponents.

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Step 3.3.1.2.1

Multiply $8$ by $9$ .

$6 \frac{72 (2 \cdot 8 + 1)}{6}$

Step 3.3.1.2.2

Multiply $2$ by $8$ .

$6 \frac{72 (16 + 1)}{6}$

Step 3.3.1.3

Add $16$ and $1$ .

$6 \frac{72 \cdot 17}{6}$

Step 3.3.2

Reduce the expression by cancelling the common factors.

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Step 3.3.2.1

Multiply $72$ by $17$ .

$6 (\frac{1224}{6})$

Step 3.3.2.2

Cancel the common factor of $6$ .

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Step 3.3.2.2.1

Cancel the common factor.

$6 (\frac{1224}{6})$

Step 3.3.2.2.2

Rewrite the expression.

$1224$

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Simplify.

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Step 4.3.1

Add $8$ and $1$ .

$7 \frac{8 \cdot 9}{2}$

Step 4.3.2

Multiply $8$ by $9$ .

$7 (\frac{72}{2})$

Step 4.3.3

Divide $72$ by $2$ .

$7 \cdot 36$

Step 4.3.4

Multiply $7$ by $36$ .

$252$

Step 5

Add the results of the summations.

$1224 + 252$

Step 6

Add $1224$ and $252$ .

$1476$