Calculus Examples

$\sum_{i = 1}^{7} a + 2 i$

Step 1

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

$\sum_{k = 1}^{7} a + 2 i = \sum_{k = 1}^{7} a + 2 \sum_{k = 1}^{7} i$

Step 2

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

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Step 2.1

The formula for the summation of a constant is:

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

Step 2.2

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

$(a) (7)$

Step 2.3

Simplify.

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Step 2.3.1

Remove parentheses.

$(a) (7)$

Step 2.3.2

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

$7 a$

Step 3

Evaluate $2 \sum_{k = 1}^{7} i$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Simplify.

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Step 3.3.1

Simplify the expression.

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Step 3.3.1.1

Add $7$ and $1$ .

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

Step 3.3.1.2

Multiply $7$ by $8$ .

$2 (\frac{56}{2})$

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Cancel the common factor.

$2 (\frac{56}{2})$

Step 3.3.2.2

Rewrite the expression.

$56$

Step 4

Add the results of the summations.

$7 a + 56$