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Murphy Rodriguez
04/01/2024 · High School
Expert-Verified Answer
1) n = \(\frac { ln( F/P) } { ln( 1+ r) } \)
2) 174
UpStudy Free Solution:
Let's start with the given formula for the future value of an investment:
\[F = P ( 1 + r) ^ n\]
1) To solve for \(n\):
\[F = P ( 1 + r) ^ n\]
First, divide both sides by \(P\):
\[\frac { F} { P} = ( 1 + r) ^ n\]
Next, take the natural logarithm (ln) of both sides:
\[\ln \left ( \frac { F} { P} \right ) = \ln \left ( ( 1 + r) ^ n\right ) \]
Using the logarithm power rule \(\ln ( a^ b) = b \ln ( a) \):
\[\ln \left ( \frac { F} { P} \right ) = n \ln ( 1 + r) \]
Finally, solve for \(n\):
\[n = \frac { \ln \left ( \frac { F} { P} \right ) } { \ln ( 1 + r) } \]
2) To find the number of periods needed for an investment to double, we set \(F = 2P\):
\[2P = P ( 1 + r) ^ n\]
Divide both sides by \(P\):
\[2 = ( 1 + r) ^ n\]
Given \(r = 0.004\), we need to solve for \(n\):
\[2 = ( 1 + 0.004) ^ n\]
Take the natural logarithm of both sides:
\[\ln ( 2) = \ln ( ( 1.004) ^ n) \]
Using the logarithm power rule:
\[\ln ( 2) = n \ln ( 1.004) \]
Solve for \(n\):
\[n = \frac { \ln ( 2) } { \ln ( 1.004) } \]
Now, calculate the values:
\[\ln ( 2) \approx 0.693147\]
\[\ln ( 1.004) \approx 0.003992\]
\[n = \frac { 0.693147} { 0.003992} \]
\[n \approx 173.61\]
Rounding to the nearest whole number, \(n \approx 174\).
Therefore, it will take approximately 174 periods (years, months, etc.) for the investment to double at a rate of 0.004.
Supplemental Knowledge:
Understanding the future value of an investment is crucial in finance. The formula for the future value of an investment compounded annually is:
\[F = P ( 1 + r) ^ n\]
Where:
- \(F\) is the future value of the investment.
- \(P\) is the principal amount (initial investment).
- \(r\) is the annual interest rate (expressed as a decimal).
- \(n\) is the number of periods (years).
Key Concepts:
1. Compound Interest:
- Compound interest means that interest earned each period is added to the principal, so that in subsequent periods, interest is earned on both the initial principal and previously accumulated interest.
2. Natural Logarithm:
- The natural logarithm (\(\ln \)) is a mathematical function that helps solve equations involving exponential growth or decay.
3. Doubling Time:
- Doubling time refers to the period it takes for an investment to grow to twice its original amount under compound interest.
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