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Chapter 4: Problem 28

Find a number \(r\) such that $$ \left(1+\frac{r}{10^{75}}\right)^{\left(10^{75}\right)} \approx 4 $$

Short Answer

Expert verified
The approximate value of \(r\) is 3.

Step by step solution

01

Use the binomial expansion theorem

We will apply the binomial expansion theorem for the given expression and find the estimated value for \(r\) using its first few terms: \[ \left(1+\frac{r}{10^{75}}\right)^{\left(10^{75}\right)} \approx 1 + \binom{10^{75}}{1}\left(\frac{r}{10^{75}}\right) + \binom{10^{75}}{2}\left(\frac{r}{10^{75}}\right)^2 \]
02

Calculate the binomial coefficients

Next, we need to find the values of the binomial coefficients: \[ \binom{10^{75}}{1} = \frac{10^{75}!}{1!(10^{75}-1)!} = 10^{75} \] and \[ \binom{10^{75}}{2} = \frac{10^{75}!}{2!(10^{75}-2)!} = \frac{10^{75}(10^{75}-1)}{2} \]
03

Substitute the binomial coefficients and simplify the expression

Now, we substitute the binomial coefficients back into the expression: \[ \left(1+\frac{r}{10^{75}}\right)^{\left(10^{75}\right)} \approx 1 + 10^{75}\left(\frac{r}{10^{75}}\right) + \frac{10^{75}(10^{75}-1)}{2}\left(\frac{r}{10^{75}}\right)^2 \] Simplify the expression: \[ \left(1+\frac{r}{10^{75}}\right)^{\left(10^{75}\right)} \approx 1 + r + \frac{(10^{75}-1)r^2}{2\cdot 10^{150}} \]
04

Set the expression to be approximately equal to 4 and solve for r

Now, we want to find the value of \(r\) such that the expression is approximately equal to 4: \[ 1 + r + \frac{(10^{75}-1)r^2}{2\cdot 10^{150}} \approx 4 \] Subtract 1 from both sides and ignore the higher-order terms because they will be negligibly small compared to the linear term: \[ r \approx 3 \] So, the value of \(r\) is approximately equal to 3. Thus, when \(r \approx 3\), the given expression is approximately equal to 4: \[ \left(1+\frac{3}{10^{75}}\right)^{\left(10^{75}\right)} \approx 4 \]

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