Precalculus Examples

$s = \frac{n}{2} (2 a + (n - 1) d)$

Step 1

Rewrite the equation as $\frac{n}{2} \cdot (2 a + (n - 1) d) = s$ .

$\frac{n}{2} \cdot (2 a + (n - 1) d) = s$

Step 2

Multiply each term in $\frac{n}{2} \cdot (2 a + (n - 1) d) = s$ by $\frac{2}{n}$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{n}{2} \cdot (2 a + (n - 1) d) = s$ by $\frac{2}{n}$ .

$\frac{n}{2} \cdot (2 a + (n - 1) d) \frac{2}{n} = s \frac{2}{n}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the distributive property.

$\frac{n}{2} \cdot (2 a + n d - 1 d) \frac{2}{n} = s \frac{2}{n}$

Step 2.2.1.2

Rewrite $- 1 d$ as $- d$ .

$\frac{n}{2} \cdot (2 a + n d - d) \frac{2}{n} = s \frac{2}{n}$

Step 2.2.2

Apply the distributive property.

$(\frac{n}{2} (2 a) + \frac{n}{2} (n d) + \frac{n}{2} (- d)) \frac{2}{n} = s \frac{2}{n}$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Rewrite using the commutative property of multiplication.

$(2 \frac{n}{2} a + \frac{n}{2} (n d) + \frac{n}{2} (- d)) \frac{2}{n} = s \frac{2}{n}$

Step 2.2.3.2

Multiply $\frac{n}{2} (n d)$ .

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Step 2.2.3.2.1

Combine $n$ and $\frac{n}{2}$ .

$(2 \frac{n}{2} a + \frac{n \cdot n}{2} d + \frac{n}{2} (- d)) \frac{2}{n} = s \frac{2}{n}$

Step 2.2.3.2.2

Raise $n$ to the power of $1$ .

$(2 \frac{n}{2} a + \frac{n^{1} n}{2} d + \frac{n}{2} (- d)) \frac{2}{n} = s \frac{2}{n}$

Step 2.2.3.2.3

Raise $n$ to the power of $1$ .

$(2 \frac{n}{2} a + \frac{n^{1} n^{1}}{2} d + \frac{n}{2} (- d)) \frac{2}{n} = s \frac{2}{n}$

Step 2.2.3.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(2 \frac{n}{2} a + \frac{n^{1 + 1}}{2} d + \frac{n}{2} (- d)) \frac{2}{n} = s \frac{2}{n}$

Step 2.2.3.2.5

Add $1$ and $1$ .

$(2 \frac{n}{2} a + \frac{n^{2}}{2} d + \frac{n}{2} (- d)) \frac{2}{n} = s \frac{2}{n}$

Step 2.2.3.2.6

Combine $\frac{n^{2}}{2}$ and $d$ .

$(2 \frac{n}{2} a + \frac{n^{2} d}{2} + \frac{n}{2} (- d)) \frac{2}{n} = s \frac{2}{n}$

Step 2.2.3.3

Rewrite using the commutative property of multiplication.

$(2 \frac{n}{2} a + \frac{n^{2} d}{2} - \frac{n}{2} d) \frac{2}{n} = s \frac{2}{n}$

Step 2.2.4

Simplify terms.

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

Cancel the common factor of $2$ .

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Step 2.2.4.1.1.1

Cancel the common factor.

$(2 \frac{n}{2} a + \frac{n^{2} d}{2} - \frac{n}{2} d) \frac{2}{n} = s \frac{2}{n}$

Step 2.2.4.1.1.2

Rewrite the expression.

$(n a + \frac{n^{2} d}{2} - \frac{n}{2} d) \frac{2}{n} = s \frac{2}{n}$

Step 2.2.4.1.2

Combine $d$ and $\frac{n}{2}$ .

$(n a + \frac{n^{2} d}{2} - \frac{d n}{2}) \frac{2}{n} = s \frac{2}{n}$

Step 2.2.4.2

Multiply $n a + \frac{n^{2} d}{2} - \frac{d n}{2}$ by $\frac{2}{n}$ .

$\frac{(n a + \frac{n^{2} d}{2} - \frac{d n}{2}) \cdot 2}{n} = s \frac{2}{n}$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Factor $n$ out of $n a + \frac{n^{2} d}{2} - \frac{d n}{2}$ .

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Step 2.2.5.1.1

Factor $n$ out of $n a$ .

$\frac{(n (a) + \frac{n^{2} d}{2} - \frac{d n}{2}) \cdot 2}{n} = s \frac{2}{n}$

Step 2.2.5.1.2

Factor $n$ out of $\frac{n^{2} d}{2}$ .

$\frac{(n (a) + n \frac{n d}{2} - \frac{d n}{2}) \cdot 2}{n} = s \frac{2}{n}$

Step 2.2.5.1.3

Factor $n$ out of $- \frac{d n}{2}$ .

$\frac{(n (a) + n \frac{n d}{2} + n (- \frac{d}{2})) \cdot 2}{n} = s \frac{2}{n}$

Step 2.2.5.1.4

Factor $n$ out of $n (a) + n \frac{n d}{2}$ .

$\frac{(n (a + \frac{n d}{2}) + n (- \frac{d}{2})) \cdot 2}{n} = s \frac{2}{n}$

Step 2.2.5.1.5

Factor $n$ out of $n (a + \frac{n d}{2}) + n (- \frac{d}{2})$ .

$\frac{n (a + \frac{n d}{2} - \frac{d}{2}) \cdot 2}{n} = s \frac{2}{n}$

Step 2.2.5.2

To write $a$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{n (a \cdot \frac{2}{2} + \frac{n d}{2} - \frac{d}{2}) \cdot 2}{n} = s \frac{2}{n}$

Step 2.2.5.3

Combine $a$ and $\frac{2}{2}$ .

$\frac{n (\frac{a \cdot 2}{2} + \frac{n d}{2} - \frac{d}{2}) \cdot 2}{n} = s \frac{2}{n}$

Step 2.2.5.4

Combine the numerators over the common denominator.

$\frac{n (\frac{a \cdot 2 + n d}{2} - \frac{d}{2}) \cdot 2}{n} = s \frac{2}{n}$

Step 2.2.5.5

Combine the numerators over the common denominator.

$\frac{n \frac{a \cdot 2 + n d - d}{2} \cdot 2}{n} = s \frac{2}{n}$

Step 2.2.5.6

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

$\frac{n \frac{2 a + n d - d}{2} \cdot 2}{n} = s \frac{2}{n}$

Step 2.2.5.7

Combine exponents.

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Step 2.2.5.7.1

Combine $n$ and $\frac{2 a + n d - d}{2}$ .

$\frac{\frac{n (2 a + n d - d)}{2} \cdot 2}{n} = s \frac{2}{n}$

Step 2.2.5.7.2

Combine $\frac{n (2 a + n d - d)}{2}$ and $2$ .

$\frac{\frac{n (2 a + n d - d) \cdot 2}{2}}{n} = s \frac{2}{n}$

Step 2.2.5.8

Reduce the expression by cancelling the common factors.

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Step 2.2.5.8.1

Reduce the expression $\frac{n (2 a + n d - d) \cdot 2}{2}$ by cancelling the common factors.

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Step 2.2.5.8.1.1

Cancel the common factor.

$\frac{\frac{n (2 a + n d - d) \cdot 2}{2}}{n} = s \frac{2}{n}$

Step 2.2.5.8.1.2

Rewrite the expression.

$\frac{\frac{n (2 a + n d - d)}{1}}{n} = s \frac{2}{n}$

Step 2.2.5.8.2

Divide $n (2 a + n d - d)$ by $1$ .

$\frac{n (2 a + n d - d)}{n} = s \frac{2}{n}$

Step 2.2.6

Cancel the common factor of $n$ .

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Step 2.2.6.1

Cancel the common factor.

$\frac{n (2 a + n d - d)}{n} = s \frac{2}{n}$

Step 2.2.6.2

Divide $2 a + n d - d$ by $1$ .

$2 a + n d - d = s \frac{2}{n}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Combine $s$ and $\frac{2}{n}$ .

$2 a + n d - d = \frac{s \cdot 2}{n}$

Step 2.3.2

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

$2 a + n d - d = \frac{2 s}{n}$

Step 3

Move all terms not containing $a$ to the right side of the equation.

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Step 3.1

Subtract $n d$ from both sides of the equation.

$2 a - d = \frac{2 s}{n} - n d$

Step 3.2

Add $d$ to both sides of the equation.

$2 a = \frac{2 s}{n} - n d + d$

Step 4

Divide each term in $2 a = \frac{2 s}{n} - n d + d$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 a = \frac{2 s}{n} - n d + d$ by $2$ .

$\frac{2 a}{2} = \frac{\frac{2 s}{n}}{2} + \frac{- n d}{2} + \frac{d}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{\frac{2 s}{n}}{2} + \frac{- n d}{2} + \frac{d}{2}$

Step 4.2.1.2

Divide $a$ by $1$ .

$a = \frac{\frac{2 s}{n}}{2} + \frac{- n d}{2} + \frac{d}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$a = \frac{2 s}{n} \cdot \frac{1}{2} + \frac{- n d}{2} + \frac{d}{2}$

Step 4.3.1.2

Cancel the common factor of $2$ .

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Step 4.3.1.2.1

Factor $2$ out of $2 s$ .

$a = \frac{2 (s)}{n} \cdot \frac{1}{2} + \frac{- n d}{2} + \frac{d}{2}$

Step 4.3.1.2.2

Cancel the common factor.

$a = \frac{2 s}{n} \cdot \frac{1}{2} + \frac{- n d}{2} + \frac{d}{2}$

Step 4.3.1.2.3

Rewrite the expression.

$a = \frac{s}{n} + \frac{- n d}{2} + \frac{d}{2}$

Step 4.3.1.3

Move the negative in front of the fraction.

$a = \frac{s}{n} - \frac{n d}{2} + \frac{d}{2}$