Chapter 13: Problem 58
Find \(x\) so that \(2 x, 3 x+2,\) and \(5 x+3\) are consecutive terms of an arithmetic sequence.
Over 30 million students worldwide already upgrade their learning with Vaia!
These are the key concepts you need to understand to accurately answer the question.
All the tools & learning materials you need for study success - in one app.
Get started for free
Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 1+4+4^{2}+\cdots+4^{n-1}=\frac{1}{3}\left(4^{n}-1\right) $$
You are interviewing for a job and receive two offers for a five-year contract: A: \(\$ 40,000\) to start, with guaranteed annual increases of \(6 \%\) for the first 5 years B: \(\$ 44,000\) to start, with guaranteed annual increases of \(3 \%\) for the first 5 years Which offer is better if your goal is to be making as much as possible after 5 years? Which is better if your goal is to make as much money as possible over the contract (5 years)?
Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 1 \cdot 2+2 \cdot 3+3 \cdot 4+\cdots+n(n+1)=\frac{1}{3} n(n+1)(n+2) $$
Reflections in a Mirror A highly reflective mirror reflects \(95 \%\) of the light that falls on it. In a light box having walls made of the mirror, the light reflects back-and-forth between the mirrors. (a) If the original intensity of the light is \(I_{0}\) before it falls on a mirror, write the \(n\) th term of the sequence that describes the intensity of the light after \(n\) reflections. (b) How many reflections are needed to reduce the light intensity by at least \(98 \% ?\)
Suppose that, throughout the U.S. economy, individuals spend \(90 \%\) of every additional dollar that they earn. Economists would say that an individual's marginal propensity to consume is \(0.90 .\) For example, if Jane earns an additional dollar, she will spend \(0.9(1)=\$ 0.90\) of it. The individual who earns \(\$ 0.90\) (from Jane) will spend \(90 \%\) of it, or \(\$ 0.81 .\) This process of spending continues and results in an infinite geometric series as follows: $$1,0.90,0.90^{2}, 0.90^{3}, 0.90^{4}, \ldots$$ The sum of this infinite geometric series is called the multiplier. What is the multiplier if individuals spend \(90 \%\) of every additional dollar that they earn?
What do you think about this solution?
We value your feedback to improve our textbook solutions.
