Recursive equation (?)

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fable147

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Hi, I've been trying to solve this question for a few days now, but I'm not even sure if my approach is right. I would appreciate any help. Thank you!

The question:

Combinatorics

Applicants for a position are offered a starting salary of 50,000 coins and at the beginning of each calendar year their salary will be increased by 20% of the previous year's salary. (That is, for example, in the fifth year, they receive an added 20% of the salary from the fourth year.) In addition, each year the employee receives a bonus of 2,000 coins for each year already worked. (for example, in the fifth year, this added bonus will be 8,000 coins)

a) Create a recursive equation for the salary of the candidate in the n-th year of employment in this company.

b) Find the solution of the equation and determine from it what would be the salary of the candidate in the twentieth year?

How I tried to solve it:
(but it didn't work because with the equation the result salaries are negative which is obviously not right)
IMG_20211203_122256.jpg
 
I do not see what you were trying do with all those calculations.

In your very first line, you got off track. Try working out a few terms of the sequence.

[math]S_1 = 50,000\\ S_2 = 50,000 * 1.2 + 2000\\ S_3 = (50,000 * 1.2) * 1.2 + 2000 + 2000[/math]
Do you see a pattern emerging?
 
JeffM said:
I do not see what you were trying do with all those calculations.

In your very first line, you got off track. Try working out a few terms of the sequence.

[math]S_1 = 50,000\\ S_2 = 50,000 * 1.2 + 2000\\ S_3 = (50,000 * 1.2) * 1.2 + 2000 + 2000[/math]
Do you see a pattern emerging?
Click to expand...
I think [imath]S_3 = (50,000*1,2 + 2000) * 1,2 + 4000[/imath], because the previous year salary was S_2 and the bonus is now 4,000.
So I created the general formula [imath]S_n = S_{n-1}* 1,2 + 2000*(n-1)[/imath] and then tried to solve it as inhomogeneous recursion.
 
JeffM said:
[math]S_1 = 50,000\\ S_2 = 50,000 * 1.2 + 2000\\ S_3 = (50,000 * 1.2) * 1.2 + 2000 + 2000[/math]
Click to expand...
The whole [imath]S_2[/imath] should be multiplied by 1.2:
[math]S_3 = (50,000 * 1.2+2000) * 1.2 + 2000[/math]
 
I found the mistake! I accidentaly forgot a zero in the final equation.[math]50000 = 1.2^n*A - 10000n - 50000[/math] so [math]A = 110,000/1.2[/math] and the final form is [math]a_n = 110,000*1.2^n - 10,000n - 50,000[/math]Anyways. Thank you for help!
 
JeffM said:
That is far from clear from the wording. A bonus is not usually considered part of salary.
Click to expand...
Good point. I assumed the bonus was included based on the first line of the OP's solution.
 
lev888 said:
Good point. I assumed the bonus was included based on the first line of the OP's solution.
Click to expand...
The word “coins” is odd. I suspect that what we are seeing is a translation of the original problem. If so, we may easily miss a nuance.
 
JeffM said:
That is far from clear from the wording. A bonus is not usually considered part of salary.
Click to expand...
IRS considers as part of salary - you have to pay usual FICA , etc. taxes
 
JeffM said:
The word “coins” is odd. I suspect that what we are seeing is a translation of the original problem. If so, we may easily miss a nuance.
Click to expand...
The corporate finance of course would agree with Jeff - so that bonus is not factored in while calculating raise given.
 
JeffM said:
The word “coins” is odd. I suspect that what we are seeing is a translation of the original problem. If so, we may easily miss a nuance.
Click to expand...
May be the payments are made by bit (non-existent) coins - just look at 20% raise every year.
 
Let's look at it this way.
Suppose you have a yearly salary of $60,000. Your boss gives you a $5,000 bonus at the end of the year. Next year you get a 3% raise. Would you expect your new yearly salary to be $60,000*1.03 or $65,000*1.03? I think the former is right.
 
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