Basic Math Examples

$93000 = 20000 ((\frac{{(1 + 0.02)}^{n + 1} - 1}{0.02}) - 1)$

Step 1

Rewrite the equation as $20000 (\frac{{(1 + 0.02)}^{n + 1} - 1}{0.02} - 1) = 93000$ .

$20000 (\frac{{(1 + 0.02)}^{n + 1} - 1}{0.02} - 1) = 93000$

Step 2

Add $1$ and $0.02$ .

$20000 (\frac{{1.02}^{n + 1} - 1}{0.02} - 1) = 93000$

Step 3

Divide each term in $20000 (\frac{{1.02}^{n + 1} - 1}{0.02} - 1) = 93000$ by $20000$ and simplify.

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Step 3.1

Divide each term in $20000 (\frac{{1.02}^{n + 1} - 1}{0.02} - 1) = 93000$ by $20000$ .

$\frac{20000 (\frac{{1.02}^{n + 1} - 1}{0.02} - 1)}{20000} = \frac{93000}{20000}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $20000$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{20000 (\frac{{1.02}^{n + 1} - 1}{0.02} - 1)}{20000} = \frac{93000}{20000}$

Step 3.2.1.2

Divide $\frac{{1.02}^{n + 1} - 1}{0.02} - 1$ by $1$ .

$\frac{{1.02}^{n + 1} - 1}{0.02} - 1 = \frac{93000}{20000}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Multiply by $1$ .

$\frac{1 \cdot {1.02}^{n + 1} - 1}{0.02} - 1 = \frac{93000}{20000}$

Step 3.2.2.2

Rewrite $- 1$ as $1 (- 1)$ .

$\frac{1 \cdot {1.02}^{n + 1} + 1 (- 1)}{0.02} - 1 = \frac{93000}{20000}$

Step 3.2.2.3

Factor $1$ out of $1 \cdot {1.02}^{n + 1} + 1 (- 1)$ .

$\frac{1 \cdot ({1.02}^{n + 1} - 1)}{0.02} - 1 = \frac{93000}{20000}$

Step 3.2.2.4

Factor $0.02$ out of $0.02$ .

$\frac{1 \cdot ({1.02}^{n + 1} - 1)}{0.02 (1)} - 1 = \frac{93000}{20000}$

Step 3.2.2.5

Separate fractions.

$\frac{1}{0.02} \cdot \frac{{1.02}^{n + 1} - 1}{1} - 1 = \frac{93000}{20000}$

Step 3.2.2.6

Divide $1$ by $0.02$ .

$50 \frac{{1.02}^{n + 1} - 1}{1} - 1 = \frac{93000}{20000}$

Step 3.2.2.7

Divide ${1.02}^{n + 1} - 1$ by $1$ .

$50 ({1.02}^{n + 1} - 1) - 1 = \frac{93000}{20000}$

Step 3.2.2.8

Apply the distributive property.

$50 \cdot {1.02}^{n + 1} + 50 \cdot - 1 - 1 = \frac{93000}{20000}$

Step 3.2.2.9

Multiply $50$ by $- 1$ .

$50 \cdot {1.02}^{n + 1} - 50 - 1 = \frac{93000}{20000}$

Step 3.2.3

Subtract $1$ from $- 50$ .

$50 \cdot {1.02}^{n + 1} - 51 = \frac{93000}{20000}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $93000$ and $20000$ .

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Step 3.3.1.1

Factor $1000$ out of $93000$ .

$50 \cdot {1.02}^{n + 1} - 51 = \frac{1000 (93)}{20000}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $1000$ out of $20000$ .

$50 \cdot {1.02}^{n + 1} - 51 = \frac{1000 \cdot 93}{1000 \cdot 20}$

Step 3.3.1.2.2

Cancel the common factor.

$50 \cdot {1.02}^{n + 1} - 51 = \frac{1000 \cdot 93}{1000 \cdot 20}$

Step 3.3.1.2.3

Rewrite the expression.

$50 \cdot {1.02}^{n + 1} - 51 = \frac{93}{20}$

Step 4

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

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Step 4.1

Add $51$ to both sides of the equation.

$50 \cdot {1.02}^{n + 1} = \frac{93}{20} + 51$

Step 4.2

To write $51$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$50 \cdot {1.02}^{n + 1} = \frac{93}{20} + 51 \cdot \frac{20}{20}$

Step 4.3

Combine $51$ and $\frac{20}{20}$ .

$50 \cdot {1.02}^{n + 1} = \frac{93}{20} + \frac{51 \cdot 20}{20}$

Step 4.4

Combine the numerators over the common denominator.

$50 \cdot {1.02}^{n + 1} = \frac{93 + 51 \cdot 20}{20}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $51$ by $20$ .

$50 \cdot {1.02}^{n + 1} = \frac{93 + 1020}{20}$

Step 4.5.2

Add $93$ and $1020$ .

$50 \cdot {1.02}^{n + 1} = \frac{1113}{20}$

Step 5

Divide each term in $50 \cdot {1.02}^{n + 1} = \frac{1113}{20}$ by $50$ and simplify.

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Step 5.1

Divide each term in $50 \cdot {1.02}^{n + 1} = \frac{1113}{20}$ by $50$ .

$\frac{50 \cdot {1.02}^{n + 1}}{50} = \frac{\frac{1113}{20}}{50}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $50$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{50 \cdot {1.02}^{n + 1}}{50} = \frac{\frac{1113}{20}}{50}$

Step 5.2.1.2

Divide ${1.02}^{n + 1}$ by $1$ .

${1.02}^{n + 1} = \frac{\frac{1113}{20}}{50}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

${1.02}^{n + 1} = \frac{1113}{20} \cdot \frac{1}{50}$

Step 5.3.2

Multiply $\frac{1113}{20} \cdot \frac{1}{50}$ .

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Step 5.3.2.1

Multiply $\frac{1113}{20}$ by $\frac{1}{50}$ .

${1.02}^{n + 1} = \frac{1113}{20 \cdot 50}$

Step 5.3.2.2

Multiply $20$ by $50$ .

${1.02}^{n + 1} = \frac{1113}{1000}$

Step 6

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({1.02}^{n + 1}) = \ln (\frac{1113}{1000})$

Step 7

Expand $\ln ({1.02}^{n + 1})$ by moving $n + 1$ outside the logarithm.

$(n + 1) \ln (1.02) = \ln (\frac{1113}{1000})$

Step 8

Simplify the left side.

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Step 8.1

Simplify $(n + 1) \ln (1.02)$ .

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Step 8.1.1

Apply the distributive property.

$n \ln (1.02) + 1 \ln (1.02) = \ln (\frac{1113}{1000})$

Step 8.1.2

Multiply $\ln (1.02)$ by $1$ .

$n \ln (1.02) + \ln (1.02) = \ln (\frac{1113}{1000})$

Step 9

Move all the terms containing a logarithm to the left side of the equation.

$n \ln (1.02) + \ln (1.02) - \ln (\frac{1113}{1000}) = 0$

Step 10

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$n \ln (1.02) + \ln (\frac{1.02}{\frac{1113}{1000}}) = 0$

Step 11

Simplify each term.

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Step 11.1

Multiply the numerator by the reciprocal of the denominator.

$n \ln (1.02) + \ln (1.02 (\frac{1000}{1113})) = 0$

Step 11.2

Multiply $1.02 (\frac{1000}{1113})$ .

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Step 11.2.1

Combine $1.02$ and $\frac{1000}{1113}$ .

$n \ln (1.02) + \ln (\frac{1.02 \cdot 1000}{1113}) = 0$

Step 11.2.2

Multiply $1.02$ by $1000$ .

$n \ln (1.02) + \ln (\frac{1020}{1113}) = 0$

Step 11.3

Cancel the common factor of $1020$ and $1113$ .

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Step 11.3.1

Factor $3$ out of $1020$ .

$n \ln (1.02) + \ln (\frac{3 (340)}{1113}) = 0$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Factor $3$ out of $1113$ .

$n \ln (1.02) + \ln (\frac{3 \cdot 340}{3 \cdot 371}) = 0$

Step 11.3.2.2

Cancel the common factor.

$n \ln (1.02) + \ln (\frac{3 \cdot 340}{3 \cdot 371}) = 0$

Step 11.3.2.3

Rewrite the expression.

$n \ln (1.02) + \ln (\frac{340}{371}) = 0$

Step 12

Subtract $\ln (\frac{340}{371})$ from both sides of the equation.

$n \ln (1.02) = - \ln (\frac{340}{371})$

Step 13

Divide each term in $n \ln (1.02) = - \ln (\frac{340}{371})$ by $\ln (1.02)$ and simplify.

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Step 13.1

Divide each term in $n \ln (1.02) = - \ln (\frac{340}{371})$ by $\ln (1.02)$ .

$\frac{n \ln (1.02)}{\ln (1.02)} = \frac{- \ln (\frac{340}{371})}{\ln (1.02)}$

Step 13.2

Simplify the left side.

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Step 13.2.1

Cancel the common factor of $\ln (1.02)$ .

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Step 13.2.1.1

Cancel the common factor.

$\frac{n \ln (1.02)}{\ln (1.02)} = \frac{- \ln (\frac{340}{371})}{\ln (1.02)}$

Step 13.2.1.2

Divide $n$ by $1$ .

$n = \frac{- \ln (\frac{340}{371})}{\ln (1.02)}$

Step 13.3

Simplify the right side.

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Step 13.3.1

Move the negative in front of the fraction.

$n = - \frac{\ln (\frac{340}{371})}{\ln (1.02)}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{\ln (\frac{340}{371})}{\ln (1.02)}$

Decimal Form:

$n = 4.40630648 \dots$