prove that : 2^n + 2^n-1 / 2^n+1 - 2^n = 3/2
To prove the given equation: 2^n + 2^(n-1) / 2^(n+1) - 2^n = 3/2, we can simplify both sides of the equation and show that they are equal.
Let's start by simplifying the left side of the equation:
2^n + 2^(n-1) / 2^(n+1) - 2^n
First, let's focus on the denominator, which is 2^(n+1). We can rewrite this as 2^n * 2^1, which is equal to 2^n * 2.
Now, let's rewrite the expression:
2^n + 2^(n-1) / 2^n * 2 - 2^n
Next, let's simplify the numerator. Since 2^n and 2^(n-1) have a common base of 2, we can combine them:
2^n + 2^(n-1) = 2^n * (1 + 1/2) = 2^n * (2/2 + 1/2) = 2^n * 3/2
Now, let's rewrite the expression:
2^n * 3/2 / 2^n * 2 - 2^n
Next, let's simplify the denominator by combining like terms:
2^n * 3/2 / 2^(n+1) - 2^n
Now, let's simplify the denominator further:
2^n * 3/2 / 2^n * 2 - 2^n = 2^n * 3/2 / 2^n * 2 - 2^n * 1
Next, let's cancel out the common terms in the numerator and the denominator:
(3/2) / (2 * 1)
Now, let's simplify the expression:
(3/2) / 2 = (3/2) * (1/2) = 3/4
Therefore, we have proven that 2^n + 2^(n-1) / 2^(n+1) - 2^n = 3/4, not 3/2.
It appears there might be a mistake in the given equation. Double-check the original equation or provide any additional information if needed.
( 2 ^ n + 2 ^ ( n - 1 ) ) / ( 2 ^ ( n + 1 ) - 2 ^ n ) Multiply both sides by ( 2 ^ ( n + 1 ) - 2 ^ n )
________________________________________
( 2 ^ n ) * 2 ^ ( n + 1 ) = 2 ^ ( 2 n + 1 )
2 ^ ( n - 1 ) * (2 ^ ( n + 1 ) ) = 2 ^ 2 n
2 ^ ( 2 n + 1 ) + 2 ^ 2 n = 3 * 4 ^ n
2 ^ ( 2 n ) * 2 ^ ( 2 n ) = ( 2 ^ ( 2 n ) ) ^ 2 = 2 ^ ( 4 n )
________________________________________
( 2 ^ n + 2 ^ ( n - 1 ) ) / ( 2 ^ ( n + 1 ) - 2 ^ n ) =
2 ^ ( 2 n + 1 ) + 2 ^ 2 n / ( 2 ^ ( 2 n ) ) ^ 2 =
3 * 4 ^ n / ( 2 * 4 ^ n ) = 3 / 2