SOLUTION: Hello and Thank You for your help. I know the answewr to my problem is 10 years. I kept subtracting to get the answer. I don't know how to write an equation to solve the problem

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Question 199771: Hello and Thank You for your help.
I know the answewr to my problem is 10 years. I kept subtracting to get the answer. I don't know how to write an equation to solve the problem. The question is
Bill will put aside $100 per month this year for retiremenmt. Next year he will put away $200 per month the third year $300 per month. Each year will increase by $100 per month. How many years will Bill save to reach $66,000?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
You basically have the sequence:
12*100,
12*200,
12*300,
12*400,
12*500, ...


note: I'm multiplying by 12 since he deposits money 12 times a year.


and you're adding up the terms to get:
12*100+12*200=12*300,
12*100+12*200+12*300=12*600,
12*100+12*200+12*300+12*400=12*1000,
12*100+12*200+12*300+12*400+12*500=12*1500, ...



and you want to know which partial sum will be equal to 66,000.


So you basically want to know when 12*100+12*200+12*300+12*400+12*500+...=66000


We can factor out the GCF 12*100 to get: 12*100(1+2+3+4+5+....)=66000


and we're left with a very familiar series in the parenthesis.


It turns out that (this is a well known formula).


So the equation then becomes: 12%2A100%28%28n%28n%2B1%29%29%2F2%29=66000


where "n" is the number of years.


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12%2A100%28%28n%28n%2B1%29%29%2F2%29=66000 Start with the given equation.


100%28%28n%28n%2B1%29%29%2F2%29=66000%2F12 Divide both sides by 12.


100%28%28n%28n%2B1%29%29%2F2%29=5500 Reduce


100%28n%28n%2B1%29%29%2F2=5500%2F100 Divide both sides by 100.


%28n%28n%2B1%29%29%2F2=55 Reduce.


n%28n%2B1%29=55%2A2 Multiply both sides by 2.


n%28n%2B1%29=110 Multiply


n%5E2%2Bn=110 Distribute


n%5E2%2Bn-110=0 Subtract 110 from both sides.


Notice we have a quadratic in the form of An%5E2%2BBn%2BC where A=1, B=1, and C=-110


Let's use the quadratic formula to solve for "n":


n+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


n+=+%28-%281%29+%2B-+sqrt%28+%281%29%5E2-4%281%29%28-110%29+%29%29%2F%282%281%29%29 Plug in A=1, B=1, and C=-110


n+=+%28-1+%2B-+sqrt%28+1-4%281%29%28-110%29+%29%29%2F%282%281%29%29 Square 1 to get 1.


n+=+%28-1+%2B-+sqrt%28+1--440+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-110%29 to get -440


n+=+%28-1+%2B-+sqrt%28+1%2B440+%29%29%2F%282%281%29%29 Rewrite sqrt%281--440%29 as sqrt%281%2B440%29


n+=+%28-1+%2B-+sqrt%28+441+%29%29%2F%282%281%29%29 Add 1 to 440 to get 441


n+=+%28-1+%2B-+sqrt%28+441+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


n+=+%28-1+%2B-+21%29%2F%282%29 Take the square root of 441 to get 21.


n+=+%28-1+%2B+21%29%2F%282%29 or n+=+%28-1+-+21%29%2F%282%29 Break up the expression.


n+=+%2820%29%2F%282%29 or n+=++%28-22%29%2F%282%29 Combine like terms.


n+=+10 or n+=+-11 Simplify.


So the possible solutions are n+=+10 or n+=+-11

Recall that the value of "n" is the number of years. Since you cannot have a negative year, this excludes n+=+-11


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Answer:


So the answer is n+=+10 which means that it will take 10 years to save up to $66,000.


If you have any questions, email me at jim_thompson5910@hotmail.com