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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?

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1 Answer

up vote 1 down vote accepted

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

Can you finish it from there?

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Got it. thanks. –  yiyi Sep 27 '12 at 4:22
    
@MaoYiyi: Great! You’re welcome. –  Brian M. Scott Sep 27 '12 at 4:25
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