Solve for n 25000=400((1-(1.0025)^(-n))/0.0025)(1.0025)

Math
25000=400(1-(1.0025)-n0.0025)(1.0025)
Rewrite the equation as 400(1-(1.0025)-n0.0025)⋅(1.0025)=25000.
400(1-(1.0025)-n0.0025)⋅(1.0025)=25000
Simplify.
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Combine 400 and 1-(1.0025)-n0.0025.
400(1-(1.0025)-n)0.0025⋅1.0025=25000
Combine 400(1-(1.0025)-n)0.0025 and 1.0025.
400(1-(1.0025)-n)⋅1.00250.0025=25000
Multiply 1.0025 by 400.
401(1-(1.0025)-n)0.0025=25000
401(1-(1.0025)-n)0.0025=25000
Divide each term by 401 and simplify.
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Divide each term in 401(1-(1.0025)-n)0.0025=25000 by 401.
401(1-(1.0025)-n)0.0025401=25000401
Cancel the common factor of 401.
400-400⋅1.0025-n=25000401
Cancel the common factor of 25000 and 401.
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Rewrite 25000 as 1(25000).
400-400⋅1.0025-n=1(25000)401
Cancel the common factors.
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Rewrite 401 as 1(401).
400-400⋅1.0025-n=1⋅250001⋅401
Cancel the common factor.
400-400⋅1.0025-n=1⋅250001⋅401
Rewrite the expression.
400-400⋅1.0025-n=25000401
400-400⋅1.0025-n=25000401
400-400⋅1.0025-n=25000401
400-400⋅1.0025-n=25000401
Move all terms not containing n to the right side of the equation.
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Subtract 400 from both sides of the equation.
-400⋅1.0025-n=25000401-400
To write -400 as a fraction with a common denominator, multiply by 401401.
-400⋅1.0025-n=25000401-400⋅401401
Combine -400 and 401401.
-400⋅1.0025-n=25000401+-400⋅401401
Combine the numerators over the common denominator.
-400⋅1.0025-n=25000-400⋅401401
Simplify the numerator.
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Multiply -400 by 401.
-400⋅1.0025-n=25000-160400401
Subtract 160400 from 25000.
-400⋅1.0025-n=-135400401
-400⋅1.0025-n=-135400401
Move the negative in front of the fraction.
-400⋅1.0025-n=-135400401
-400⋅1.0025-n=-135400401
Divide each term by -400 and simplify.
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Divide each term in -400⋅1.0025-n=-135400401 by -400.
-400⋅1.0025-n-400=-135400401⋅(-1400)
Cancel the common factor of -400.
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Cancel the common factor.
-400⋅1.0025-n-400=-135400401⋅(-1400)
Divide 1.0025-n by 1.
1.0025-n=-135400401⋅(-1400)
1.0025-n=-135400401⋅(-1400)
Simplify -135400401⋅(-1400).
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Cancel the common factor of 200.
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Move the leading negative in -135400401 into the numerator.
1.0025-n=-135400401⋅(-1400)
Move the leading negative in -1400 into the numerator.
1.0025-n=-135400401⋅-1400
Factor 200 out of -135400.
1.0025-n=200(-677)401⋅-1400
Factor 200 out of 400.
1.0025-n=200⋅-677401⋅-1200⋅2
Cancel the common factor.
1.0025-n=200⋅-677401⋅-1200⋅2
Rewrite the expression.
1.0025-n=-677401⋅-12
1.0025-n=-677401⋅-12
Multiply -677401 and -12.
1.0025-n=-677⋅-1401⋅2
Multiply.
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Multiply -677 by -1.
1.0025-n=677401⋅2
Multiply 401 by 2.
1.0025-n=677802
1.0025-n=677802
1.0025-n=677802
1.0025-n=677802
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
ln(1.0025-n)=ln(677802)
Expand ln(1.0025-n) by moving -n outside the logarithm.
-nln(1.0025)=ln(677802)
Divide each term by -1ln(1.0025) and simplify.
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Divide each term in -nln(1.0025)=ln(677802) by -1ln(1.0025).
-nln(1.0025)-1ln(1.0025)=ln(677802)-1ln(1.0025)
Simplify -nln(1.0025)-1ln(1.0025).
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Dividing two negative values results in a positive value.
nln(1.0025)ln(1.0025)=ln(677802)-1ln(1.0025)
Cancel the common factor of ln(1.0025).
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Cancel the common factor.
nln(1.0025)ln(1.0025)=ln(677802)-1ln(1.0025)
Divide n by 1.
n=ln(677802)-1ln(1.0025)
n=ln(677802)-1ln(1.0025)
n=ln(677802)-1ln(1.0025)
Move the negative in front of the fraction.
n=-ln(677802)ln(1.0025)
n=-ln(677802)ln(1.0025)
The result can be shown in multiple forms.
Exact Form:
n=-ln(677802)ln(1.0025)
Decimal Form:
n=67.85961739…
Solve for n 25000=400((1-(1.0025)^(-n))/0.0025)(1.0025)

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