Calculus Examples

$\sum_{n = 3}^{8} n + 25$

Step 1

Split the summation to make the starting value of $n$ equal to $1$ .

$\sum_{n = 3}^{8} n + 25 = \sum_{n = 1}^{8} n + 25 - \sum_{n = 1}^{2} n + 25$

Step 2

Evaluate $\sum_{n = 1}^{8} n + 25$ .

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Step 2.1

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

$\sum_{n = 1}^{8} n + 25 = \sum_{n = 1}^{8} n + \sum_{n = 1}^{8} 25$

Step 2.2

Evaluate $\sum_{n = 1}^{8} n$ .

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Step 2.2.1

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

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

Step 2.2.2

Substitute the values into the formula.

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

Step 2.2.3

Simplify.

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Step 2.2.3.1

Cancel the common factor of $8$ and $2$ .

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Step 2.2.3.1.1

Factor $2$ out of $8 (8 + 1)$ .

$\frac{2 (4 (8 + 1))}{2}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $2$ out of $2$ .

$\frac{2 (4 (8 + 1))}{2 (1)}$

Step 2.2.3.1.2.2

Cancel the common factor.

$\frac{2 (4 (8 + 1))}{2 \cdot 1}$

Step 2.2.3.1.2.3

Rewrite the expression.

$\frac{4 (8 + 1)}{1}$

Step 2.2.3.1.2.4

Divide $4 (8 + 1)$ by $1$ .

$4 (8 + 1)$

Step 2.2.3.2

Simplify the expression.

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Step 2.2.3.2.1

Add $8$ and $1$ .

$4 \cdot 9$

Step 2.2.3.2.2

Multiply $4$ by $9$ .

$36$

Step 2.3

Evaluate $\sum_{n = 1}^{8} 25$ .

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Step 2.3.1

The formula for the summation of a constant is:

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

Step 2.3.2

Substitute the values into the formula.

$(25) (8)$

Step 2.3.3

Multiply $25$ by $8$ .

$200$

Step 2.4

Add the results of the summations.

$36 + 200$

Step 2.5

Add $36$ and $200$ .

$236$

Step 3

Evaluate $\sum_{n = 1}^{2} n + 25$ .

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Step 3.1

Expand the series for each value of $n$ .

$1 + 25 + 2 + 25$

Step 3.2

Simplify.

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Step 3.2.1

Add $1$ and $25$ .

$26 + 2 + 25$

Step 3.2.2

Add $2$ and $25$ .

$26 + 27$

Step 3.2.3

Add $26$ and $27$ .

$53$

Step 4

Replace the summations with the values found.

$236 - 53$

Step 5

Subtract $53$ from $236$ .

$183$