Finite Math Examples

$98000 = 29000 (\frac{1 - {(1.067)}^{- n}}{0.067})$

Step 1

Rewrite the equation as $29000 (\frac{1 - {1.067}^{- n}}{0.067}) = 98000$ .

$29000 (\frac{1 - {1.067}^{- n}}{0.067}) = 98000$

Step 2

Simplify $29000 (\frac{1 - {1.067}^{- n}}{0.067})$ .

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Step 2.1

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

$29000 \frac{1 (1) - {1.067}^{- n}}{0.067} = 98000$

Step 2.2

Factor $1$ out of $- {1.067}^{- n}$ .

$29000 \frac{1 (1) + 1 (- {1.067}^{- n})}{0.067} = 98000$

Step 2.3

Factor $1$ out of $1 (1) + 1 (- {1.067}^{- n})$ .

$29000 \frac{1 (1 - {1.067}^{- n})}{0.067} = 98000$

Step 2.4

Factor $0.067$ out of $0.067$ .

$29000 \frac{1 (1 - {1.067}^{- n})}{0.067 (1)} = 98000$

Step 2.5

Separate fractions.

$29000 (\frac{1}{0.067} \cdot \frac{1 - {1.067}^{- n}}{1}) = 98000$

Step 2.6

Simplify the expression.

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Step 2.6.1

Divide $1$ by $0.067$ .

$29000 (14.92537313 \frac{1 - {1.067}^{- n}}{1}) = 98000$

Step 2.6.2

Divide $1 - {1.067}^{- n}$ by $1$ .

$29000 (14.92537313 (1 - {1.067}^{- n})) = 98000$

Step 2.7

Apply the distributive property.

$29000 (14.92537313 \cdot 1 + 14.92537313 (- {1.067}^{- n})) = 98000$

Step 2.8

Multiply.

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Step 2.8.1

Multiply $14.92537313$ by $1$ .

$29000 (14.92537313 + 14.92537313 (- {1.067}^{- n})) = 98000$

Step 2.8.2

Multiply $- 1$ by $14.92537313$ .

$29000 (14.92537313 - 14.92537313 \cdot {1.067}^{- n}) = 98000$

Step 2.9

Apply the distributive property.

$29000 \cdot 14.92537313 + 29000 (- 14.92537313 \cdot {1.067}^{- n}) = 98000$

Step 2.10

Multiply.

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Step 2.10.1

Multiply $29000$ by $14.92537313$ .

$432835.82089552 + 29000 (- 14.92537313 \cdot {1.067}^{- n}) = 98000$

Step 2.10.2

Multiply $- 14.92537313$ by $29000$ .

$432835.82089552 - 432835.82089552 \cdot {1.067}^{- n} = 98000$

Step 3

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

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Step 3.1

Subtract $432835.82089552$ from both sides of the equation.

$- 432835.82089552 \cdot {1.067}^{- n} = 98000 - 432835.82089552$

Step 3.2

Subtract $432835.82089552$ from $98000$ .

$- 432835.82089552 \cdot {1.067}^{- n} = - 334835.82089552$

Step 4

Divide each term in $- 432835.82089552 \cdot {1.067}^{- n} = - 334835.82089552$ by $- 432835.82089552$ and simplify.

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Step 4.1

Divide each term in $- 432835.82089552 \cdot {1.067}^{- n} = - 334835.82089552$ by $- 432835.82089552$ .

$\frac{- 432835.82089552 \cdot {1.067}^{- n}}{- 432835.82089552} = \frac{- 334835.82089552}{- 432835.82089552}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 432835.82089552$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 432835.82089552 \cdot {1.067}^{- n}}{- 432835.82089552} = \frac{- 334835.82089552}{- 432835.82089552}$

Step 4.2.1.2

Divide ${1.067}^{- n}$ by $1$ .

${1.067}^{- n} = \frac{- 334835.82089552}{- 432835.82089552}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 334835.82089552$ by $- 432835.82089552$ .

${1.067}^{- n} = 0.7735862$

Step 5

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

$\ln ({1.067}^{- n}) = \ln (0.7735862)$

Step 6

Expand $\ln ({1.067}^{- n})$ by moving $- n$ outside the logarithm.

$- n \ln (1.067) = \ln (0.7735862)$

Step 7

Divide each term in $- n \ln (1.067) = \ln (0.7735862)$ by $- \ln (1.067)$ and simplify.

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Step 7.1

Divide each term in $- n \ln (1.067) = \ln (0.7735862)$ by $- \ln (1.067)$ .

$\frac{- n \ln (1.067)}{- \ln (1.067)} = \frac{\ln (0.7735862)}{- \ln (1.067)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Dividing two negative values results in a positive value.

$\frac{n \ln (1.067)}{\ln (1.067)} = \frac{\ln (0.7735862)}{- \ln (1.067)}$

Step 7.2.2

Cancel the common factor of $\ln (1.067)$ .

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Step 7.2.2.1

Cancel the common factor.

$\frac{n \ln (1.067)}{\ln (1.067)} = \frac{\ln (0.7735862)}{- \ln (1.067)}$

Step 7.2.2.2

Divide $n$ by $1$ .

$n = \frac{\ln (0.7735862)}{- \ln (1.067)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$n = - \frac{\ln (0.7735862)}{\ln (1.067)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{\ln (0.7735862)}{\ln (1.067)}$

Decimal Form:

$n = 3.95858621 \dots$