MHB HUGGIE BUGGIE's question at Yahoo Answers regarding calculating a sum

[MarkFL]

Here is the question:

Math Homework HELP! PLEASE!?


If your starting salary were $50,000 and you received a 4% increase at the end of every year for 15 years, what would be the total amount, in dollars, you would have earned over the first 16 years that you worked?

I have no idea how to get the answer to this please help me!

I have to round my answer to the nearest dollar

I have posted a link there to this thread so the OP can see my work.

 

[MarkFL]

Hello HUGGIE BUGGIE!,

Your salary $S$ for the $k$th year in dollars is:

$$S_k=50000\left(\frac{26}{25} \right)^{k-1}$$

And hence, the sum total $T$ of the money earned during the first $n$ years is:

$$T_n=50000\sum_{k=1}^{n}\left(\frac{26}{25} \right)^{k-1}=50000\sum_{k=0}^{n-1}\left(\frac{26}{25} \right)^{k}$$

Using the formula:

$$\sum_{k=0}^{n}r^k=\frac{r^{n+1}-1}{r-1}$$

we may write:

$$T_n=50000\left(\frac{\left(\dfrac{26}{25} \right)^n-1}{\frac{26}{25}-1} \right)=1250000\left(\left(\frac{26}{25} \right)^n-1 \right)$$

Thus, the money earned during the first 16 years is:

$$T_{16}=1250000\left(\left(\frac{26}{25} \right)^{16}-1 \right)=\frac{325210856544670578706416}{298023223876953125}\approx1091227$$