Solving for coefficients?

maria.pilar

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Joined
Mar 3, 2015
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Hi everyone! Could you please help me find the "smart" and "short" solution to this problem? Step by step would be useful. I can solve it the long way, but I am looking for a quicker path. Thanks!

I buy a car for $8,000. I will use my monthly wages to pay for it.
At the end of this month I will earn $1,000.
My wages will increase by 5% every month.
In how many months will my car be completely paid off?

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So the long way would be to add the monthly wages until the total is at least $8,000:
1,000 + 1,000*1.05 + 1,000*1.05*1.05 + 1,000*1.05*1.05*1.05.... => 8,000
OR
1,000 + 1,000*1.05 + 1,000*1.05^2 + 1,000*1.05^3... => 8,000

The answer is 7 months, which add to 8,142.87 - but it is long and tedious to add these results until we reach 8,000.

I can see that we are dealing with the same base, 1.05.
I tried simplifying the equation but my method does not work because n is unknown:
1,000 (1 + 1.05 + 1.05^2 + 1.05^3...) => 8,000

I also tried another manner, which did not work either when applying it to different cases:
1,000x + 1,000*1.05^x => 8,000
1,000x + 1,000*xln1.05 => 8,000

All your help is appreciated :)
 
This is called a geometric progression.

let Sn = the total money you earn after n months

Sn = 1000 + 1000 * 1.05 + 1000 * 1.052... + 1000 * 1.05n -1
1.05 * Sn =1000 * 1.05 + 1000 * 1.052... + 1000 * 1.05n

therefore 1.05 * Sn -Sn = 1000 * 1.05n - 1000 because the terms in the middle cancel out.

1.05 * Sn -Sn = 0.05Sn
therefore Sn = 20 * (1000 * 1.05n - 1000)

ill let you work out the rest.
Hope this helped!
 
Thanks!!

Thanks a lot! I've been breaking my head, dreaming of it! I had never heard of geometric progressions. It's more complex than expected. I have to spend some time processing it. I appreciate your time and guidance.

Thank you thank you!
:D
 
Thank you Denis for the example! Helps in order to understand and ultimately memorize the formula :D
I appreciate your time.
 
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