Prove that-2n-2n-1-2n-1-2n-3-2

To Prove: 2^n+2^n 1/2^n+1 2^n=3/2On simplifying the LHS, we get =2^n+2^n 1/2^n+1 2^nTaking out the common term 2^n from numerator and denominator, we get, =(2^n ...

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Byju's Answer

Standard IX

Mathematics

Fractional Exponent

Prove that:2n...

Question

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

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Solution

To Prove: $\frac{2^{n} + 2^{n - 1}}{2^{n + 1} - 2^{n}} = \frac{3}{2}$
On simplifying the LHS, we get
$= \frac{2^{n} + 2^{n - 1}}{2^{n + 1} - 2^{n}}$
Taking out the common term $2^{n}$ from numerator and denominator,
we get,
$= \frac{(2^{n} + 2^{n} 2^{- 1})}{(2^{n} 2^{1} - 2^{n})} = \frac{2^{n} (1 + 1 \times 2^{- 1})}{2^{n} (2^{1} - 1)}$
Cancel the common $2^{n}$ from both numerator and denominator,
we get,
$= \frac{1 + 2^{- 1}}{2 - 1}$ [Using $a^{- n} = \frac{1}{a^{n}}$ ]
$= \frac{1 + \frac{1}{2}}{1} = \frac{3}{2}$ [Taking LCM]
Hence proved.

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Prove that:
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