Calculus Examples

$\sum_{i = 1}^{3} \frac{a i + b}{2 a}$

Step 1

Expand the series for each value of $i$ .

$\frac{a \cdot 1 + b}{2 a} + \frac{a \cdot 2 + b}{2 a} + \frac{a \cdot 3 + b}{2 a}$

Step 2

Simplify.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{a \cdot 1 + b + a \cdot 2 + b + a \cdot 3 + b}{2 a}$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $a$ by $1$ .

$\frac{a + b + a \cdot 2 + b + a \cdot 3 + b}{2 a}$

Step 2.2.2

Move $2$ to the left of $a$ .

$\frac{a + b + 2 \cdot a + b + a \cdot 3 + b}{2 a}$

Step 2.2.3

Move $3$ to the left of $a$ .

$\frac{a + b + 2 a + b + 3 a + b}{2 a}$

Step 2.3

Add $a$ and $2 a$ .

$\frac{3 a + b + b + 3 a + b}{2 a}$

Step 2.4

Add $3 a$ and $3 a$ .

$\frac{6 a + b + b + b}{2 a}$

Step 2.5

Add $b$ and $b$ .

$\frac{6 a + 2 b + b}{2 a}$

Step 2.6

Add $2 b$ and $b$ .

$\frac{6 a + 3 b}{2 a}$

Step 2.7

Factor $3$ out of $6 a + 3 b$ .

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Step 2.7.1

Factor $3$ out of $6 a$ .

$\frac{3 (2 a) + 3 b}{2 a}$

Step 2.7.2

Factor $3$ out of $3 b$ .

$\frac{3 (2 a) + 3 (b)}{2 a}$

Step 2.7.3

Factor $3$ out of $3 (2 a) + 3 (b)$ .

$\frac{3 (2 a + b)}{2 a}$