Solving a Math Problem on Income with 5% Increases over 25 Years

[darkmagic]

How do you solve this?
There is this kind of question in our test and I don't know how will I do it.

You're working in a company. Your starting income is 5000. Every year, the income will increase by 5%. What is your total income on your 25th year in the company?

 

[HallsofIvy]

This is a question, not a tutorial so I am moving out of "Learning Materials" to "Precalculus Homework and School Work".

You startwith income at 5000 and it increases by 5% each year.

Okay, the first year your increases by "5% of 5000"= .05(5000)= 250 so your income the second year is 5250= 5000+ (.05)5000= (1.05)(5000). At the end of that year it increases by "5% of 5250"= .05(5250)= 262.50 and the third year your income is 5512.50= 5250+ (.05)5250= (1.05)(5250)= 1.05(1.05(5000)). The reason I wrote it out like that is because neither you nor I want to do that 24 times! (During your 25th year, your income will have increased 24 times.) You should be able to see what is happening: each year your income is multiplied by 1.05. After 24 years, that initial 5000 is multiplied by 1.05 24 times: $(1.05)^{24}(5000)$.

 

[Дьявол]

$$a_1=5000$$

$$a_2=a_1+a_1*\frac{5}{100}=a_1*1.05$$

$$a_3=a_1*1.05 + a_1*1.05*0.05=a_1*1.05(1 + 0.05)=a_1*1.05*1.05$$

$$a_4=a_1*1.05*1.05*1.05$$

$$.......$$

$$a_{n+1}=a_1*(1.05)^{n}$$

So a₂₅=5000*(1.05)²⁴

Regards.

 

[darkmagic]

Is 1.05 constant?

 

[Дьявол]

Is 1.05 constant?

Yep, its a constant.

 

[HallsofIvy]

It's certainly not going to change!

 

[darkmagic]

ok. where is that 1.05 come from?

 

[Дьявол]

ok. where is that 1.05 come from?

If you see my way of solving the problem, you'll spot that

$$a_2=a_1+a_1*\frac{5}{100}=a_1(1+\frac{5}{100})=a_1\frac{100+5}{100}=a_1\frac{105}{100}=a_1*1.05$$

See now where it comes from?

 

[HallsofIvy]

ok. where is that 1.05 come from?

I thought I had said that. Each year, your income is the same as the previous year (100%) and an additional 5% raise. Together that means your income each year is 105% of the previous year's. 105% is the same as 1.05.

 

[calcatraz.com]

If it helps anyone, you can work this sort of thing out on Calcatraz, my new online calculator.

Just enter it as: http://www.calcatraz.com/calc?c=%245000+raised+by+5%25+in+year+25" , substituting your own values.