What is the answer to ((2^n))/(n+1)=51.2 ?

The answer to ((2^n))/(n+1)=51.2 is n=9,n=-(\W_{0}(-\frac{ln(2))/(102.4))+ln(2)}{ln(2)}

Solve for n: ((2^n))/(n+1)=51.2

The solution is n=9,n=-(\W_{0}(-\frac{ln(2))/(102.4))+ln(2)}{ln(2)}

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Popular >

((2^n))/(n+1)=51.2

Solution

\frac{( 2 ^{n} )}{n + 1} = 51.2

Solution

n = 9, n = - \frac{W _{0} ( - \frac{l n ( 2 )}{102.4} ) + ln ( 2 )}{ln ( 2 )}

Decimal Notation

n = 9, n = - 0.99016 \dots

Solution steps

\frac{( 2 ^{n} )}{n + 1} = 51.2

Prepare

\frac{( 2 ^{n} )}{n + 1} = 51.2

for Lambert form:

1 = 51.2 (n + 1) e^{- l n (2) n}

Rewrite the equation with

(- n - 1) ln (2) = u

and

n = - \frac{u + ln ( 2 )}{ln ( 2 )}

1 = 51.2 ((- \frac{u + ln ( 2 )}{ln ( 2 )}) + 1) e^{- l n (2) (- \frac{u + l n ( 2 )}{l n ( 2 )})}

Simplify

1 = 51.2 e^{u + l n (2)} (- \frac{u + ln ( 2 )}{ln ( 2 )} + 1)

Rewrite

1 = 51.2 e^{u + l n (2)} (- \frac{u + ln ( 2 )}{ln ( 2 )} + 1)

in Lambert form:

e^{u} u = - \frac{ln ( 2 )}{102.4}

Solve

e^{u} u = - \frac{ln ( 2 )}{102.4} : u = - 10 ln (2), u = W_{0} (- \frac{ln ( 2 )}{102.4})

u = - 10 ln (2), u = W_{0} (- \frac{ln ( 2 )}{102.4})

Substitute back

u = (- n - 1) ln (2),

solve for

n

Solve

(- n - 1) ln (2) = - 10 ln (2) : n = 9

Solve

(- n - 1) ln (2) = W_{0} (- \frac{ln ( 2 )}{102.4}) : n = - \frac{W _{0} ( - \frac{l n ( 2 )}{102.4} ) + ln ( 2 )}{ln ( 2 )}

n = 9, n = - \frac{W _{0} ( - \frac{l n ( 2 )}{102.4} ) + ln ( 2 )}{ln ( 2 )}

Verify Solutions:

n = 9

True

, n = - \frac{W _{0} ( - \frac{l n ( 2 )}{102.4} ) + ln ( 2 )}{ln ( 2 )}

True

The solutions are

n = 9, n = - \frac{W _{0} ( - \frac{l n ( 2 )}{102.4} ) + ln ( 2 )}{ln ( 2 )}

Verify Solutions:

n = 9

True

, n = - \frac{W _{0} ( - \frac{l n ( 2 )}{102.4} ) + ln ( 2 )}{ln ( 2 )}

True

The solutions are

n = 9, n = - \frac{W _{0} ( - \frac{l n ( 2 )}{102.4} ) + ln ( 2 )}{ln ( 2 )}

Popular Examples

8^{x-9}=(1/2)^{x+1}

8^{x - 9} = (\frac{1}{2})^{x + 1}

3^{5x}=12

3^{5 x} = 12

9^{3x-12}=(3)^{2x}

9^{3 x - 12} = (3)^{2 x}

3^{2x+1}=27^{x+2}

3^{2 x + 1} = 2 7^{x + 2}

(1/9)^{x-5}=27

(\frac{1}{9})^{x - 5} = 27

Frequently Asked Questions (FAQ)

What is the answer to ((2^n))/(n+1)=51.2 ?
The answer to ((2^n))/(n+1)=51.2 is n=9,n=-(\W_{0}(-\frac{ln(2))/(102.4))+ln(2)}{ln(2)}
Solve for n: ((2^n))/(n+1)=51.2
The solution is n=9,n=-(\W_{0}(-\frac{ln(2))/(102.4))+ln(2)}{ln(2)}