Solve for n:
1000= 50 x(1.00125)^n + 10 x(1.00125^n - 1/1.00125 - 1)
Please show your steps.
Thanks
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⋅(1.00125n−11.00125−1)|a=1.001251000=50⋅an+10⋅(an−1a−1)|:10100=5⋅an+(an−1a−1)100=5⋅an+an−1a−1100=6⋅an−1a−1|+1101=6⋅an−1a6⋅an−1a=101|+1a6⋅an=101+1a|:6an=101+1a6|log10()log10(an)=log10(101+1a6)n⋅log10(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
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.
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⋅(1.00125n−11.00125−1)|a=1.001251000=50⋅an+10⋅(an−1a−1)|:10100=5⋅an+(an−1a−1)100=5⋅an+an−1a−1100=6⋅an−1a−1|+1101=6⋅an−1a6⋅an−1a=101|+1a6⋅an=101+1a|:6an=101+1a6|log10()log10(an)=log10(101+1a6)n⋅log10(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