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Solve for n:

 

1000= 50 x(1.00125)^n + 10 x(1.00125^n - 1/1.00125 - 1)

 

Please show your steps.

 

Thanks

 May 10, 2016

Best Answer 

 #2
avatar+26397 
+5

Solve for n:

1000= 50 x(1.00125)^n + 10 x(1.00125^n - 1/1.00125 - 1)

Please show your steps.

 

Let a = 1.00125

 

1000=50(1.00125)n+10(1.00125n11.001251)|a=1.001251000=50an+10(an1a1)|:10100=5an+(an1a1)100=5an+an1a1100=6an1a1|+1101=6an1a6an1a=101|+1a6an=101+1a|:6an=101+1a6|log10()log10(an)=log10(101+1a6)nlog10(a)=log10(101+1a6)|log10(a)n=log10(101+1a6)log10(a)|a=1.00125n=log10(101+11.001256)log10(1.00125)n=log10(101+0.998751560556)log10(1.00125)n=log10(101.9987515616)log10(1.00125)n=log10(16.9997919268)log10(1.00125)n=1.230443605750.00054252909n=2267.97718911

 

laugh

 May 11, 2016
edited by heureka  May 11, 2016
edited by heureka  May 11, 2016
 #1
avatar
+5

Solve for n over the real numbers:
1000 = 50 1.00125^n+10 (1.00125^n-1.99875)

 

50 1.00125^n+10 (1.00125^n-1.99875) = 2^(1-5 n) 25^(1-n) 801^n+10 ((801/800)^n-1601/801):
1000 = 2^(1-5 n) 25^(1-n) 801^n+10 ((801/800)^n-1601/801)

 

1000 = 2^(1-5 n) 25^(1-n) 801^n+10 ((801/800)^n-1601/801) is equivalent to 2^(1-5 n) 25^(1-n) 801^n+10 ((801/800)^n-1601/801) = 1000:
2^(1-5 n) 25^(1-n) 801^n+10 ((801/800)^n-1601/801) = 1000

 

2^(1-5 n) 25^(1-n) 801^n+10 ((801/800)^n-1601/801)  =  -16010/801+2^(1-5 n) 5^(1-2 n) 801^n+2^(1-5 n) 25^(1-n) 801^n:
-16010/801+2^(1-5 n) 5^(1-2 n) 801^n+2^(1-5 n) 25^(1-n) 801^n = 1000

 

2^(1-5 n) 5^(1-2 n) 801^n = e^(log(2^(1-5 n))) e^(log(5^(1-2 n))) e^(log(801^n)) = e^((1-5 n) log(2)) e^((1-2 n) log(5)) e^(n log(801)) = exp((1-5 n) log(2)+(1-2 n) log(5)+n log(801)) and 2^(1-5 n) 25^(1-n) 801^n = e^(log(2^(1-5 n))) e^(log(25^(1-n))) e^(log(801^n)) = e^((1-5 n) log(2)) e^((1-n) log(25)) e^(n log(801)) = exp((1-5 n) log(2)+(1-n) log(25)+n log(801)):
-16010/801+exp(log(2) (1-5 n)+log(5) (1-2 n)+log(801) n)+exp(log(2) (1-5 n)+log(25) (1-n)+log(801) n) = 1000

 

Simplify and substitute x = exp((1-5 n) log(2)+(1-2 n) log(5)+n log(801)):
 -16010/801+exp((1-5 n) log(2)+(1-2 n) log(5)+n log(801))+exp((1-5 n) log(2)+(1-n) log(25)+n log(801))  =  6 e^((1-5 n) log(2)+(1-2 n) log(5)+n log(801))-16010/801  =  6 x-16010/801  =  1000:
6 x-16010/801 = 1000

 

Add 16010/801 to both sides:
6 x = 817010/801

 

Divide both sides by 6:
x = 408505/2403

 

Substitute back for x = exp((1-5 n) log(2)+(1-2 n) log(5)+n log(801)):
exp(log(2) (1-5 n)+log(5) (1-2 n)+log(801) n) = 408505/2403

 

Take the natural logarithm of both sides:
log(2) (1-5 n)+log(5) (1-2 n)+log(801) n = log(408505/2403)

 

Expand and collect in terms of n:
(-5 log(2)-2 log(5)+log(801)) n+log(2)+log(5) = log(408505/2403)

 

Subtract log(2)+log(5) from both sides:
(log(801)+(-5 log(2)-2 log(5))) n = log(408505/2403)+(-log(2)-log(5))

 

Divide both sides by -5 log(2)-2 log(5)+log(801):
Answer: |  n = (-log(2)-log(5)+log(408505/2403))/(-5 log(2)-2 log(5)+log(801))=~2268

 

IT DOES BALANCE, BUT THERE MUST BE A SIMPLER WAY!!. MAYBE CPhill has one.

 May 10, 2016
 #2
avatar+26397 
+5
Best Answer

Solve for n:

1000= 50 x(1.00125)^n + 10 x(1.00125^n - 1/1.00125 - 1)

Please show your steps.

 

Let a = 1.00125

 

1000=50(1.00125)n+10(1.00125n11.001251)|a=1.001251000=50an+10(an1a1)|:10100=5an+(an1a1)100=5an+an1a1100=6an1a1|+1101=6an1a6an1a=101|+1a6an=101+1a|:6an=101+1a6|log10()log10(an)=log10(101+1a6)nlog10(a)=log10(101+1a6)|log10(a)n=log10(101+1a6)log10(a)|a=1.00125n=log10(101+11.001256)log10(1.00125)n=log10(101+0.998751560556)log10(1.00125)n=log10(101.9987515616)log10(1.00125)n=log10(16.9997919268)log10(1.00125)n=1.230443605750.00054252909n=2267.97718911

 

laugh

heureka May 11, 2016
edited by heureka  May 11, 2016
edited by heureka  May 11, 2016

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