Question

Simplify : 2 ^n + 2 ^n-1 ÷ 2^n+1 - 2^n.

Answer 1

(2^n+2^n-1)/(2^n+1-2^n)

Take 2^n common

∴ 2^n(1+2^-1)/2^n(2-1)

=> (1+2^-1)/1

= 1+2^-1

=1+1/2

=3/2

Answer 2

$\bf \red{\frac{ {2}^{n} + {2}^{n - 1} }{ {2}^{n + 1} - {2}^{n} } = \frac{3}{2}}$

Given:

• $\frac{ {2}^{n} + {2}^{n - 1} }{ {2}^{n + 1} - {2}^{n} }$

To find:

• Simplify.

Solution:

Concept to be used:

1. $\bf {a}^{-n} = \frac{1}{a^{n} }$
2. $\bf {a}^{(n+m)} = {a}^{n} {a}^{m}$

Step 1:

Rewrite the expression.

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

or

Take common

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

or

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

Step 2:

Simplify the mathematical expression.

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

or

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

or

$= \frac{3}{2}$

Thus,

$\bf \frac{ {2}^{n} + {2}^{n - 1} }{ {2}^{n + 1} - {2}^{n} } = \frac{3}{2}$

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