Calculus Examples

$\sum_{n = 5}^{10} n$

Step 1

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

$\sum_{n = 5}^{10} n = \sum_{n = 1}^{10} n - \sum_{n = 1}^{4} n$

Step 2

Evaluate $\sum_{n = 1}^{10} n$ .

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

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

Substitute the values into the formula.

$\frac{10 (10 + 1)}{2}$

Simplify.

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Cancel the common factor of $10$ and $2$ .

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Factor $2$ out of $10 (10 + 1)$ .

$\frac{2 (5 (10 + 1))}{2}$

Cancel the common factors.

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Factor $2$ out of $2$ .

$\frac{2 (5 (10 + 1))}{2 (1)}$

Cancel the common factor.

$\frac{2 (5 (10 + 1))}{2 \cdot 1}$

Rewrite the expression.

$\frac{5 (10 + 1)}{1}$

Divide $5 (10 + 1)$ by $1$ .

$5 (10 + 1)$

Simplify the expression.

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Add $10$ and $1$ .

$5 \cdot 11$

Multiply $5$ by $11$ .

$55$

Step 3

Evaluate $\sum_{n = 1}^{4} n$ .

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Expand the series for each value of $n$ .

$1 + 2 + 3 + 4$

Simplify.

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Add $1$ and $2$ .

$3 + 3 + 4$

Add $3$ and $3$ .

$6 + 4$

Add $6$ and $4$ .

$10$

Step 4

Replace the summations with the values found.

$55 - 10$

Step 5

Subtract $10$ from $55$ .

$45$