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

MarkFL

Staff member
Here is the question:

If your starting salary were $50,000 and you received a 4% increase at the end of every year for 15 years, what would be the total amount, in dollars, you would have earned over the first 16 years that you worked? I have no idea how to get the answer to this please help me! I have to round my answer to the nearest dollar I have posted a link there to this thread so the OP can see my work. MarkFL Administrator Staff member Hello HUGGIE BUGGIE!!!, Your salary$S$for the$k$th year in dollars is: $$\displaystyle S_k=50000\left(\frac{26}{25} \right)^{k-1}$$ And hence, the sum total$T$of the money earned during the first$n\$ years is:

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

Using the formula:

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

we may write:

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

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

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