Prove that:$ \frac{2^{n}+2^{n-1}}{2^{n+1}-2^{n}}=\frac{3}{2} $

Given: 

\( \frac{2^{n}+2^{n-1}}{2^{n+1}-2^{n}}=\frac{3}{2} \)

To do: 

We have to prove that \( \frac{2^{n}+2^{n-1}}{2^{n+1}-2^{n}}=\frac{3}{2} \).

Solution:

We know that,

$(a^{m})^{n}=a^{m n}$

$a^{m} \times a^{n}=a^{m+n}$

$a^{m} \div a^{n}=a^{m-n}$

$a^{0}=1$

Therefore,

LHS $=\frac{2^{n}+2^{n-1}}{2^{n+1}-2^{n}}$

$=\frac{2^{n}+2^{n} \times 2^{-1}}{2^{n} \times 2^{1}-2^{n}}$

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

$=\frac{1+\frac{1}{2}}{2-1}$

$=\frac{\frac{3}{2}}{1}$

$=\frac{3}{2}$

$=$ RHS

Hence proved.    

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Updated on: 10-Oct-2022

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