Basic Math Examples

$25000 = 400 (\frac{1 - {(1.0025)}^{- n}}{0.0025}) (1.0025)$

Step 1

Rewrite the equation as $400 (\frac{1 - {1.0025}^{- n}}{0.0025}) \cdot 1.0025 = 25000$ .

$400 (\frac{1 - {1.0025}^{- n}}{0.0025}) \cdot 1.0025 = 25000$

Step 2

Simplify $400 (\frac{1 - {1.0025}^{- n}}{0.0025}) \cdot 1.0025$ .

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Step 2.1

Cancel the common factor of $0.0025$ .

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Step 2.1.1

Factor $0.0025$ out of $400$ .

$0.0025 (160000) \frac{1 - {1.0025}^{- n}}{0.0025} \cdot 1.0025 = 25000$

Step 2.1.2

Cancel the common factor.

$0.0025 \cdot 160000 \frac{1 - {1.0025}^{- n}}{0.0025} \cdot 1.0025 = 25000$

Step 2.1.3

Rewrite the expression.

$160000 (1 - {1.0025}^{- n}) \cdot 1.0025 = 25000$

Step 2.2

Apply the distributive property.

$(160000 \cdot 1 + 160000 (- {1.0025}^{- n})) \cdot 1.0025 = 25000$

Step 2.3

Multiply.

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Step 2.3.1

Multiply $160000$ by $1$ .

$(160000 + 160000 (- {1.0025}^{- n})) \cdot 1.0025 = 25000$

Step 2.3.2

Multiply $- 1$ by $160000$ .

$(160000 - 160000 \cdot {1.0025}^{- n}) \cdot 1.0025 = 25000$

Step 2.4

Apply the distributive property.

$160000 \cdot 1.0025 - 160000 \cdot {1.0025}^{- n} \cdot 1.0025 = 25000$

Step 2.5

Multiply $160000$ by $1.0025$ .

$160400 - 160000 \cdot {1.0025}^{- n} \cdot 1.0025 = 25000$

Step 2.6

Multiply ${1.0025}^{- n}$ by $1.0025$ by adding the exponents.

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Step 2.6.1

Move $1.0025$ .

$160400 - 160000 \cdot (1.0025 \cdot {1.0025}^{- n}) = 25000$

Step 2.6.2

Multiply $1.0025$ by ${1.0025}^{- n}$ .

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Step 2.6.2.1

Raise $1.0025$ to the power of $1$ .

$160400 - 160000 \cdot ({1.0025}^{1} \cdot {1.0025}^{- n}) = 25000$

Step 2.6.2.2

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

$160400 - 160000 \cdot {1.0025}^{1 - n} = 25000$

Step 3

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

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Step 3.1

Subtract $160400$ from both sides of the equation.

$- 160000 \cdot {1.0025}^{1 - n} = 25000 - 160400$

Step 3.2

Subtract $160400$ from $25000$ .

$- 160000 \cdot {1.0025}^{1 - n} = - 135400$

Step 4

Divide each term in $- 160000 \cdot {1.0025}^{1 - n} = - 135400$ by $- 160000$ and simplify.

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Step 4.1

Divide each term in $- 160000 \cdot {1.0025}^{1 - n} = - 135400$ by $- 160000$ .

$\frac{- 160000 \cdot {1.0025}^{1 - n}}{- 160000} = \frac{- 135400}{- 160000}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 160000$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 160000 \cdot {1.0025}^{1 - n}}{- 160000} = \frac{- 135400}{- 160000}$

Step 4.2.1.2

Divide ${1.0025}^{1 - n}$ by $1$ .

${1.0025}^{1 - n} = \frac{- 135400}{- 160000}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $- 135400$ and $- 160000$ .

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Step 4.3.1.1

Factor $- 200$ out of $- 135400$ .

${1.0025}^{1 - n} = \frac{- 200 (677)}{- 160000}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $- 200$ out of $- 160000$ .

${1.0025}^{1 - n} = \frac{- 200 \cdot 677}{- 200 \cdot 800}$

Step 4.3.1.2.2

Cancel the common factor.

${1.0025}^{1 - n} = \frac{- 200 \cdot 677}{- 200 \cdot 800}$

Step 4.3.1.2.3

Rewrite the expression.

${1.0025}^{1 - n} = \frac{677}{800}$

Step 5

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

$\ln ({1.0025}^{1 - n}) = \ln (\frac{677}{800})$

Step 6

Expand $\ln ({1.0025}^{1 - n})$ by moving $1 - n$ outside the logarithm.

$(1 - n) \ln (1.0025) = \ln (\frac{677}{800})$

Step 7

Simplify the left side.

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Step 7.1

Simplify $(1 - n) \ln (1.0025)$ .

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Step 7.1.1

Apply the distributive property.

$1 \ln (1.0025) - n \ln (1.0025) = \ln (\frac{677}{800})$

Step 7.1.2

Multiply $\ln (1.0025)$ by $1$ .

$\ln (1.0025) - n \ln (1.0025) = \ln (\frac{677}{800})$

Step 8

Reorder $\ln (1.0025)$ and $- n \ln (1.0025)$ .

$- n \ln (1.0025) + \ln (1.0025) = \ln (\frac{677}{800})$

Step 9

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

$- n \ln (1.0025) + \ln (1.0025) - \ln (\frac{677}{800}) = 0$

Step 10

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

$- n \ln (1.0025) + \ln (\frac{1.0025}{\frac{677}{800}}) = 0$

Step 11

Simplify each term.

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Step 11.1

Multiply the numerator by the reciprocal of the denominator.

$- n \ln (1.0025) + \ln (1.0025 (\frac{800}{677})) = 0$

Step 11.2

Multiply $1.0025 (\frac{800}{677})$ .

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Step 11.2.1

Combine $1.0025$ and $\frac{800}{677}$ .

$- n \ln (1.0025) + \ln (\frac{1.0025 \cdot 800}{677}) = 0$

Step 11.2.2

Multiply $1.0025$ by $800$ .

$- n \ln (1.0025) + \ln (\frac{802}{677}) = 0$

Step 11.3

Cancel the common factor of $802$ and $677$ .

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Step 11.3.1

Rewrite $802$ as $1 (802)$ .

$- n \ln (1.0025) + \ln (\frac{1 (802)}{677}) = 0$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Rewrite $677$ as $1 (677)$ .

$- n \ln (1.0025) + \ln (\frac{1 \cdot 802}{1 \cdot 677}) = 0$

Step 11.3.2.2

Cancel the common factor.

$- n \ln (1.0025) + \ln (\frac{1 \cdot 802}{1 \cdot 677}) = 0$

Step 11.3.2.3

Rewrite the expression.

$- n \ln (1.0025) + \ln (\frac{802}{677}) = 0$

Step 12

Subtract $\ln (\frac{802}{677})$ from both sides of the equation.

$- n \ln (1.0025) = - \ln (\frac{802}{677})$

Step 13

Divide each term in $- n \ln (1.0025) = - \ln (\frac{802}{677})$ by $- \ln (1.0025)$ and simplify.

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Step 13.1

Divide each term in $- n \ln (1.0025) = - \ln (\frac{802}{677})$ by $- \ln (1.0025)$ .

$\frac{- n \ln (1.0025)}{- \ln (1.0025)} = \frac{- \ln (\frac{802}{677})}{- \ln (1.0025)}$

Step 13.2

Simplify the left side.

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Step 13.2.1

Dividing two negative values results in a positive value.

$\frac{n \ln (1.0025)}{\ln (1.0025)} = \frac{- \ln (\frac{802}{677})}{- \ln (1.0025)}$

Step 13.2.2

Cancel the common factor of $\ln (1.0025)$ .

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Step 13.2.2.1

Cancel the common factor.

$\frac{n \ln (1.0025)}{\ln (1.0025)} = \frac{- \ln (\frac{802}{677})}{- \ln (1.0025)}$

Step 13.2.2.2

Divide $n$ by $1$ .

$n = \frac{- \ln (\frac{802}{677})}{- \ln (1.0025)}$

Step 13.3

Simplify the right side.

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Step 13.3.1

Dividing two negative values results in a positive value.

$n = \frac{\ln (\frac{802}{677})}{\ln (1.0025)}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$n = \frac{\ln (\frac{802}{677})}{\ln (1.0025)}$

Decimal Form:

$n = 67.85961739 \dots$