The function f is defined by f(x)=41(0.28)^x. For any positive integer n, the value of f(n) is p% less than the value of f(n-1). What is the value of p?

To solve for \( p \) in the given problem, we first analyze the function \( f(x) = 41(0.28)^x \).

Given that the value of \( f(n) \) is \( p\% \) less than the value of \( f(n-1) \), we can express this relationship mathematically:

\[
f(n) = (1 - \frac{p}{100}) \times f(n-1)
\]

Substitute the function \( f(x) = 41(0.28)^x \) into the equation:

\[
41(0.28)^n = (1 - \frac{p}{100}) \times 41(0.28)^{n-1}
\]

We can cancel out the common factor of 41 from both sides:

\[
(0.28)^n = (1 - \frac{p}{100}) \times (0.28)^{n-1}
\]

Next, divide both sides by \( (0.28)^{n-1} \):

\[
0.28 = 1 - \frac{p}{100}
\]

Now, solve for \( \frac{p}{100} \):

\[
\frac{p}{100} = 1 - 0.28
\]

\[
\frac{p}{100} = 0.72
\]

Finally, multiply both sides by 100 to find \( p \):

\[
p = 72
\]

Thus, the value of \( p \) is **72%**.

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. How does the value of \( p \) change if the base of the exponential function is altered?
2. What is the value of \( p \) if \( f(x) = 50(0.5)^x \)?
3. How does the percentage decrease in \( f(n) \) relate to the base of the exponent?
4. What happens to \( p \) when the function is \( f(x) = 41(0.5)^x \)?
5. How does the constant multiplier (41) affect the value of \( p \)?
6. What is the general formula for \( p \) when \( f(x) = a(b)^x \)?
7. How does the function change when \( f(x) = 41(0.28)^{2x} \)?
8. What is the significance of the base \( 0.28 \) in the context of exponential decay?

**Tip:** Understanding exponential decay functions is crucial in fields like physics and finance, where they model processes such as radioactive decay and depreciation.

Explore how to calculate percentage decrease in an exponential decay function f(x) = 41(0.28)^x for positive integers. Learn the formula and solve for p in the equation where f(n) is p% less than f(n-1).

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