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So the question goes as follows :

"An employee has a starting salary of $20,000 and can choose from two salary options. Option 1 has a salary increased of 5% a year. Option 2 has a guaranteed increase of 1,000 dollars each year."

  1. Which option is initially more beneficial?
  2. Which option is more beneficial after 10 years?

So, I know that this is a problem of geometric and arithmetic sequence. For the first question, I wrote that Option 1 would be the better one financially because it is an exponential growth and naturally it will have more than a linear graph.

For the second question, I wrote again that Option 1 would be more beneficial after 10 years because inputting the 11th term as n in the equation I have, I got $20,000(1.05)^{10}$ which would be 32,578 dollars compared to Option 2's 32,000.

Is this work correct? I just want to be sure. Thanks!

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  • $\begingroup$ Which option is initially more beneficial? That's ambiguous unless initially is better defined. Both options give the same salary for the second year. After that, the GP wins of course. $\endgroup$ – dxiv Feb 25 '18 at 5:12
  • $\begingroup$ @dxiv Yeah, what confused me at first was how initially would be defined since I also found that they both give 21,000 dollars after the first year. I think I'll just mention that then. Thanks. $\endgroup$ – user55614 Feb 25 '18 at 5:15
  • $\begingroup$ I admit I expected a catch. That n percent of salary is initially less than fixed raise r. But 5% of 20,000 is 1000. So initially they are equal but only for the first year. The arithmetic is never better. $\endgroup$ – fleablood Feb 25 '18 at 5:59
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For the second question, what you have shown is that for $n=11$, the second option is better, but you haven't shown that this holds for all $n > 11$.

To show this, notice that after $n$ years, option $1$ would give $20,000(1+0.05)^n$ dollars, while option $2$ would give $20,000(1+0.05n)$ dollars. Now we can use Bernoulli's inequality, which shows that $(1+x)^n > 1+nx$ for all real numbers $n > 1$ and $x > -1$.

Substituting $x = 0.05$ shows that option $2$ is better for all $n > 1$.

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