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Chapter 7: Problem 12

Find the smallest integer \(n\) such that \(0.9^{n}<10^{-200}\).

Short Answer

Expert verified
The smallest integer n that satisfies the inequality \(0.9^n < 10^{-200}\) is n = 4371.

Step by step solution

01

Rewrite the inequality with the same base and apply the logarithm

On the left side, we can rewrite 0.9 as \(9 \cdot 10^{-1}\) and the right side is already in the desired form. So, the inequality becomes: \((9 \cdot 10^{-1})^n < 10^{-200}\) Now, we will take a logarithm on both sides. Let's use log base 10 since that will make the calculations a little easier: \[\log_{10} ((9 \cdot 10^{-1})^n) < \log_{10}(10^{-200})\]
02

Apply properties of logarithms and solve for n

Apply the power property of logarithms, which is \(\log_a(b^c) = c \log_a b\): \(n \log_{10} (9 \cdot 10^{-1}) < -200\) Now, we use the product property of logarithms, which is \(\log_a(bc) = \log_a b + \log_a c\): \(n (\log_{10}(9) + \log_{10}(10^{-1})) < -200\) Simplify the equation by calculating the logarithm values: \(n (0.95424 - 1) < -200\) Now we have a simple linear inequality to solve for n: \(-0.04576n < -200\) Divide both sides by -0.04576 and reverse the inequality sign \(n > \frac{-200}{-0.04576} \) Calculate the result: \(n > 4370.97\)
03

Find the smallest integer n

Since n must be an integer, the smallest integer value of n that satisfies the inequality is 4371. So, the smallest integer n such that \(0.9^n < 10^{-200}\) is n = 4371.

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Most popular questions from this chapter

Consider the sequence whose \(n^{\text {th }}\) term \(a_{n}\) is given by the indicated formula. (a) Write the sequence using the three-dot notation, giving the first four terms of the sequence. (b) Write the sequence as a recursive sequence. \(a_{n}=3(-2)^{n}\)

Consider an arithmetic sequence with first term b and difference \(d\) between consecutive terms. (a) Write the sequence using the three-dot notation, giving the first four terms of the sequence. (b) Give the \(100^{\text {th }}\) term of the sequence. \(b=7, d=3\)

In Exercises 15-24, evaluate the geometric series. \(\sum_{m=3}^{77}(-5)^{m}\)

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