Calculate Starting Salary with a 10% Annual Increase – Finite Series Question

[dragon513]

Q. If a person received a 10% salary increase each year and earned a total of $155680.05 by the beginning of the 5th year, determine the starting salary.

Well, the wording is confusing me.
It tells me to get the total by the BEGINNING of the 5th year, so is it asking for the money earned from the 1st year to 4th year?

Thank you.

 

[shmoe]

That's how I would have interpreted it, if they wanted how much he had earned at the beginning of the 1st year I would have said $0.

However, the starting salary works out to be a 'nice' number if you take the $155680.05 amount to be 5 full years of working, what I'd call the beginning of the 6th year, making me believe that we're not interpreting it as intended or the problem poser made a goof.

 

[dragon513]

The answer provided is 25500 +_+
I guess it's just a bad question.

Thank you shmoe!

 

[HallsofIvy]

This is a finite "geometric sequence" with r= 1.1. There is a specific formula for the sum. However, it's easy to work out the equation for only 5 years. Let A be the amount earned the first year. Then the amount earned the second year is A+ 0.01A= (1.1)A. The amount earned the third year is (1.1) times that:
(1.1)²A. The amount earned the fourth year is (1.1)³A.

By the beginning of the fifth year he will have earned
A+ 1.1A+ 1.1²A+ 1.1<sup>3A= A(1+ 1.1+ 1.21+ 1.331)= 1.641A= 155680.05.

If it were at the end of the fifth year or beginning of the sixth year, it would be A(1+ 1.1+ 1.21+ 1.331+ 1.4641)=5.1051A=155680.05</sup>