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."
- Which option is initially more beneficial?
- 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!
Best Answer
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 $n > 1$.