Finite Math Examples

$1002.66 = 121858 (\frac{0.00675}{1 - {(1.00675)}^{- n}})$

Step 1

Rewrite the equation as $121858 (\frac{0.00675}{1 - {1.00675}^{- n}}) = 1002.66$ .

$121858 (\frac{0.00675}{1 - {1.00675}^{- n}}) = 1002.66$

Step 2

Multiply $121858 (\frac{0.00675}{1 - {1.00675}^{- n}})$ .

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Step 2.1

Combine $121858$ and $\frac{0.00675}{1 - {1.00675}^{- n}}$ .

$\frac{121858 \cdot 0.00675}{1 - {1.00675}^{- n}} = 1002.66$

Step 2.2

Multiply $121858$ by $0.00675$ .

$\frac{822.5415}{1 - {1.00675}^{- n}} = 1002.66$

Step 3

Multiply both sides by $1 - {1.00675}^{- n}$ .

$\frac{822.5415}{1 - {1.00675}^{- n}} (1 - {1.00675}^{- n}) = 1002.66 (1 - {1.00675}^{- n})$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $1 - {1.00675}^{- n}$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{822.5415}{1 - {1.00675}^{- n}} (1 - {1.00675}^{- n}) = 1002.66 (1 - {1.00675}^{- n})$

Step 4.1.1.2

Rewrite the expression.

$822.5415 = 1002.66 (1 - {1.00675}^{- n})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $1002.66 (1 - {1.00675}^{- n})$ .

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Step 4.2.1.1

Apply the distributive property.

$822.5415 = 1002.66 \cdot 1 + 1002.66 (- {1.00675}^{- n})$

Step 4.2.1.2

Multiply.

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Step 4.2.1.2.1

Multiply $1002.66$ by $1$ .

$822.5415 = 1002.66 + 1002.66 (- {1.00675}^{- n})$

Step 4.2.1.2.2

Multiply $- 1$ by $1002.66$ .

$822.5415 = 1002.66 - 1002.66 \cdot {1.00675}^{- n}$

Step 5

Solve for $n$ .

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Step 5.1

Rewrite the equation as $1002.66 - 1002.66 \cdot {1.00675}^{- n} = 822.5415$ .

$1002.66 - 1002.66 \cdot {1.00675}^{- n} = 822.5415$

Step 5.2

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

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Step 5.2.1

Subtract $1002.66$ from both sides of the equation.

$- 1002.66 \cdot {1.00675}^{- n} = 822.5415 - 1002.66$

Step 5.2.2

Subtract $1002.66$ from $822.5415$ .

$- 1002.66 \cdot {1.00675}^{- n} = - 180.1185$

Step 5.3

Divide each term in $- 1002.66 \cdot {1.00675}^{- n} = - 180.1185$ by $- 1002.66$ and simplify.

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Step 5.3.1

Divide each term in $- 1002.66 \cdot {1.00675}^{- n} = - 180.1185$ by $- 1002.66$ .

$\frac{- 1002.66 \cdot {1.00675}^{- n}}{- 1002.66} = \frac{- 180.1185}{- 1002.66}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 1002.66$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 1002.66 \cdot {1.00675}^{- n}}{- 1002.66} = \frac{- 180.1185}{- 1002.66}$

Step 5.3.2.1.2

Divide ${1.00675}^{- n}$ by $1$ .

${1.00675}^{- n} = \frac{- 180.1185}{- 1002.66}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $- 180.1185$ by $- 1002.66$ .

${1.00675}^{- n} = 0.17964065$

Step 5.4

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

$\ln ({1.00675}^{- n}) = \ln (0.17964065)$

Step 5.5

Expand $\ln ({1.00675}^{- n})$ by moving $- n$ outside the logarithm.

$- n \ln (1.00675) = \ln (0.17964065)$

Step 5.6

Divide each term in $- n \ln (1.00675) = \ln (0.17964065)$ by $- \ln (1.00675)$ and simplify.

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Step 5.6.1

Divide each term in $- n \ln (1.00675) = \ln (0.17964065)$ by $- \ln (1.00675)$ .

$\frac{- n \ln (1.00675)}{- \ln (1.00675)} = \frac{\ln (0.17964065)}{- \ln (1.00675)}$

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Dividing two negative values results in a positive value.

$\frac{n \ln (1.00675)}{\ln (1.00675)} = \frac{\ln (0.17964065)}{- \ln (1.00675)}$

Step 5.6.2.2

Cancel the common factor of $\ln (1.00675)$ .

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Step 5.6.2.2.1

Cancel the common factor.

$\frac{n \ln (1.00675)}{\ln (1.00675)} = \frac{\ln (0.17964065)}{- \ln (1.00675)}$

Step 5.6.2.2.2

Divide $n$ by $1$ .

$n = \frac{\ln (0.17964065)}{- \ln (1.00675)}$

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Move the negative in front of the fraction.

$n = - \frac{\ln (0.17964065)}{\ln (1.00675)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{\ln (0.17964065)}{\ln (1.00675)}$

Decimal Form:

$n = 255.19769961 \dots$