Finite Math Examples

$(\frac{3600 \cdot ({1.03}^{n} - 1)}{1.03 - 1}) = 58741.34853806$

Step 1

Multiply both sides of the equation by $\frac{1.03 - 1}{3600}$ .

$\frac{1.03 - 1}{3600} \cdot \frac{3600 \cdot ({1.03}^{n} - 1)}{1.03 - 1} = \frac{1.03 - 1}{3600} \cdot 58741.34853806$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{1.03 - 1}{3600} \cdot \frac{3600 \cdot ({1.03}^{n} - 1)}{1.03 - 1}$ .

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Step 2.1.1.1

Cancel the common factor of $1.03 - 1$ .

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Step 2.1.1.1.1

Cancel the common factor.

$\frac{1.03 - 1}{3600} \cdot \frac{3600 \cdot ({1.03}^{n} - 1)}{1.03 - 1} = \frac{1.03 - 1}{3600} \cdot 58741.34853806$

Step 2.1.1.1.2

Rewrite the expression.

$\frac{1}{3600} (3600 \cdot ({1.03}^{n} - 1)) = \frac{1.03 - 1}{3600} \cdot 58741.34853806$

$\frac{1}{3600} (3600 ({1.03}^{n} - 1)) = \frac{1.03 - 1}{3600} \cdot 58741.34853806$

Step 2.1.1.2

Cancel the common factor of $3600$ .

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Step 2.1.1.2.1

Cancel the common factor.

$\frac{1}{3600} (3600 ({1.03}^{n} - 1)) = \frac{1.03 - 1}{3600} \cdot 58741.34853806$

Step 2.1.1.2.2

Rewrite the expression.

${1.03}^{n} - 1 = \frac{1.03 - 1}{3600} \cdot 58741.34853806$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{1.03 - 1}{3600} \cdot 58741.34853806$ .

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Step 2.2.1.1

Subtract $1$ from $1.03$ .

${1.03}^{n} - 1 = \frac{0.03}{3600} \cdot 58741.34853806$

Step 2.2.1.2

Divide $0.03$ by $3600$ .

${1.03}^{n} - 1 = 0.000008\overline{3} \cdot 58741.34853806$

Step 2.2.1.3

Multiply $0.000008\overline{3}$ by $58741.34853806$ .

${1.03}^{n} - 1 = 0.48951123$

Step 3

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

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Step 3.1

Add $1$ to both sides of the equation.

${1.03}^{n} = 0.48951123 + 1$

Step 3.2

Add $0.48951123$ and $1$ .

${1.03}^{n} = 1.48951123$

Step 4

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

$\ln ({1.03}^{n}) = \ln (1.48951123)$

Step 5

Expand $\ln ({1.03}^{n})$ by moving $n$ outside the logarithm.

$n \ln (1.03) = \ln (1.48951123)$

Step 6

Divide each term in $n \ln (1.03) = \ln (1.48951123)$ by $\ln (1.03)$ and simplify.

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Step 6.1

Divide each term in $n \ln (1.03) = \ln (1.48951123)$ by $\ln (1.03)$ .

$\frac{n \ln (1.03)}{\ln (1.03)} = \frac{\ln (1.48951123)}{\ln (1.03)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $\ln (1.03)$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{n \ln (1.03)}{\ln (1.03)} = \frac{\ln (1.48951123)}{\ln (1.03)}$

Step 6.2.1.2

Divide $n$ by $1$ .

$n = \frac{\ln (1.48951123)}{\ln (1.03)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$n = \frac{\ln (1.48951123)}{\ln (1.03)}$

Decimal Form:

$n = 13.47984382 \dots$