Question

The approximate value of \((1.0002)^{3000}\) is

• A. 1.2
• B. 1.4
• C. 1.6
• D. 1.8

Hint:

For small \(x\), \((1+x)^n \approx 1 + nx\).

Answer 1

The Correct Option is C

To find the approximate value of \((1.0002)^{3000}\), we can use the binomial approximation or the exponential function approximation for small values of \(x\).

First, let's use the exponential approximation: 

For small values of \(a\), the expression \((1+a)^n\) can be approximated using:

\((1 + a)^n \approx e^{na}\)

where \(e\) is the base of the natural logarithm, approximately equal to 2.718.

In our problem, \(a = 0.0002\) and \(n = 3000\). So,

\(e^{na} = e^{3000 \times 0.0002} = e^{0.6}\)

Now, calculate \(e^{0.6}\):

\(e^{0.6} \approx 1.6\)

Hence, the approximate value of \((1.0002)^{3000}\) is 1.6.

This matches the correct answer given as 1.6.

To verify, note that this is consistent because the approximation for small values of \(a\) works well here, as \(a = 0.0002\) is very small.

Therefore, the correct answer is: 1.6.