# A problem on Geometric series

Xiao Ming begins a new job at a salary of 100,000. Xiao Ming expects to receive a 5% raise every year until he retires.

• Suppose that Xiao Ming works for 35 years. Determine the total salary earned over Xiao Ming's career.
• At the end of each year, Xiao Ming's employer deposits 3% of Xiao Ming's salary (for the year just finished) into a fund earning 4% per year compounded each year. Find the value of the fund just after the final deposit at the end of Xiao Ming's 35th year of employment.

I got the first part, the second part is giving me trouble. I know that

100,000(0.03)[$1.04^{34}(1.05) + 1.04^{33}(1.05)^2 + \ldots + \left(1.04\right)^2\left(1.05\right)^{33} + \left(1.04\right)\left(1.05\right)^{34}]$

How to simplify this?

-

HINT: $$\sum_{k=1}^{34}\left(1.04^k\cdot1.05^{35-k}\right)=\sum_{k=1}^{34}\left(1.04^k\cdot1.05^{-k}\right)=1.05^{35}\sum_{k=1}^{34}\left(\frac{1.04}{1.05}\right)^k$$