Solve for N......

As Melody says, this is a very interesting and college-level advanced question in financial investments. As melody has shown, there a direct solution to it, but not many people have the skill in manipulating logs as Melody has done to arrive at the accurate solution.

One look at it, it immediately tells me what it is: It is the FV of two separate TVM formulae added together as one equation: Namely, It is the FV of $1,500 per year deposited at 7% annual rate + the FV of a fixed amount of $10,000 both for N number of years, which give a combined FV of $40,396.20.

I entered those 2 formulas into my computer and combined them. Then I entered all the known variables into the computer, and by using iteration and interpolation it came out, almost immediately with an N = 10.

And to check the result, I calculated them separately as follows:

1) $1,500 doposited @ 7% for 10 years gives a FV of =$20,724.67

2) $10,000 deposited @ 7% for 10 years gives a FV of=$19,671.51

3) Adding the two FV together:$20,724.67 + $19,671.51 =40,396.18 rounded to $40,396.2, which is spot on. CONGRATULATIONS TO MELODY!.

P.S. You should heed her advice of becoming a member, if you wish to have relatively complicated questions, such as this, answered by competent people.

Hi guest,

This is a good question, if you appreciate your answer and you will use this site again, how about becoming a members.  Members often get priority when we are answering question - especially polite members that are trying hard to learn from our answers. :)    

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-1500*[(1-(1+.07)^-N)/(.07)]+40396.2*(1+.07)^-N-10000=0

 

\(-1500*\frac{1-(1+.07)^{-N}}{0.07}+40396.2*(1+.07)^{-N}-10000=0\\ -1500*\frac{1-(1.07)^{-N}}{0.07}+\frac{40396.2}{(1.07)^{N}}-10000=0\\ \frac{-150000[1-(1.07)^{-N}]}{7}+\frac{40396.2}{(1.07)^{N}}-10000=0\\ \frac{-150000[1-(1.07)^{-N}]}{1}+\frac{7*40396.2}{(1.07)^{N}}-70000=0\\ -220000+150000(1.07)^{-N}+\frac{7*40396.2}{(1.07)^{N}}=0\\ -220000(1.07)^{N}+150000(1.07)^{-N}(1.07)^{N}+7*40396.2=0\\ -220000(1.07)^{N}+150000+282773.4=0\\ -220000(1.07)^{N}+432773.4=0\\ 220000(1.07)^{N}=432773.4\\ (1.07)^{N}=\frac{432773.4}{220000}\\ (1.07)^{N}\approx 1.967151818\\ log(1.07)^{N}\approx log(1.967151818)\\ Nlog(1.07)\approx log(1.967151818)\\ N\approx \frac{ log(1.967151818}{log(1.07)}\\ \)

 

log(1.967151818)/log(1.07) = 10.0000034615204508

 

N = 10      approximately.

 

check:

 

-1500*((1-(1+.07)^-10.0000034615204508)/(.07))+40396.2*(1+.07)^(-10.0000034615204508)-10000 = 0.0000029048699091459732438786634643045252961939322074922

 

Well, that is pretty close to zero :)))

Compare Melody's solution to this step-by-step solution by Wolfram/Alpha!. I think Melody's solution is more elegant than W/A!!. So winded!!.

 

Solve for N over the real numbers:
-10000+40396.2 1.07^(-N)-21428.6 (1-1.07^(-N)) = 0

-10000+40396.2 1.07^(-N)-21428.6 (1-1.07^(-N)) = -10000+201981 (4/107)^N 5^(2 N-1)-150000/7 (1-(100/107)^N):
-10000+201981 (4/107)^N 5^(2 N-1)-150000/7 (1-(100/107)^N) = 0

Factor 107^(-N) and constant terms from the left hand side:
1/35 107^(-N) (1413867 4^N 5^(2 N)+3 4^(N+2) 5^(2 N+6)-1100000 107^N) = 0

Multiply both sides by 35:
107^(-N) (1413867 4^N 5^(2 N)+3 4^(N+2) 5^(2 N+6)-1100000 107^N) = 0

Split into two equations:
107^(-N) = 0 or 1413867 4^N 5^(2 N)+3 4^(N+2) 5^(2 N+6)-1100000 107^N = 0

107^(-N) = 0 has no solution since for all z element R, 107^z>0:
1413867 4^N 5^(2 N)+3 4^(N+2) 5^(2 N+6)-1100000 107^N = 0

4^N 5^(2 N) = e^(log(4^N)) e^(log(5^(2 N))) = e^(N log(4)) e^((2 N) log(5)) = e^(2 N log(5)+N log(4)) and 4^(N+2) 5^(2 N+6) = e^(log(4^(N+2))) e^(log(5^(2 N+6))) = e^((N+2) log(4)) e^((2 N+6) log(5)) = e^((N+2) log(4)+(2 N+6) log(5)):
-1100000 107^N+1413867 e^(2 log(5) N+log(4) N)+3 e^(log(4) (N+2)+log(5) (2 N+6)) = 0

Divide both sides by 107^N:
2163867 107^(-N) e^((log(4)+2 log(5)) N)-1100000 = 0

Bring 2163867 107^(-N) e^(N (log(4)+2 log(5)))-1100000 together using the common denominator 107^N:
-107^(-N) (1100000 107^N-2163867 e^((log(4)+2 log(5)) N)) = 0

Multiply both sides by -1:
107^(-N) (1100000 107^N-2163867 e^((log(4)+2 log(5)) N)) = 0

Split into two equations:
107^(-N) = 0 or 1100000 107^N-2163867 e^((log(4)+2 log(5)) N) = 0

107^(-N) = 0 has no solution since for all z element R, 107^z>0:
1100000 107^N-2163867 e^((log(4)+2 log(5)) N) = 0

Divide both sides by e^(N (log(4)+2 log(5))):
1100000 107^N e^((-log(4)-2 log(5)) N)-2163867 = 0

107^N e^(N (-log(4)-2 log(5))) = e^(log(107^N)) e^(log(e^(N (-log(4)-2 log(5))))) = e^(N log(107)) e^((N (-log(4)-2 log(5))) log(e)) = e^(N log(107)+N (-log(4)-2 log(5))):
1100000 e^(log(107) N+(-log(4)-2 log(5)) N)-2163867 = 0

Add 2163867 to both sides:
1100000 e^(log(107) N+(-log(4)-2 log(5)) N) = 2163867

Divide both sides by 1100000:
e^(log(107) N+(-log(4)-2 log(5)) N) = 2163867/1100000

Take the natural logarithm of both sides:
log(107) N+(-log(4)-2 log(5)) N = log(2163867/1100000)

Expand and collect in terms of N:
(-log(4)-2 log(5)+log(107)) N = log(2163867/1100000)

Divide both sides by -log(4)-2 log(5)+log(107):
Answer: | N = (log(2163867/1100000))/(-log(4)-2 log(5)+log(107))=10