Question

David is trying to solve the expression :
\[\frac{(4)^2 \times 2^{(n+1)} - 4 \times 2^n}{(4)^2 \times 2^{(n+2)} - 2 \times 2^{(n+2)}}\]
And you help him to do the same and finally arrive at the answer with correct to one decimal which would be - (Note:- DO NOT include spaces in your answer)

Answer 1

Correct Answer: 0.5

Let's simplify the given expression step-by-step.
Given expression:
\[\frac{(4)^2 \times 2^{(n+1)} - 4 \times 2^n}{(4)^2 \times 2^{(n+2)} - 2 \times 2^{(n+2)}}\]
Step 1: Simplify the Numerator
The numerator is:
\[(4)^2 \times 2^{(n+1)} - 4 \times 2^n\]
Since \((4)^2 = 16\), the numerator becomes:
\[16 \times 2^{(n+1)} - 4 \times 2^n\]
Step 2: Simplify the Denominator
The denominator is:
\[(4)^2 \times 2^{(n+2)} - 2 \times 2^{(n+2)}\]
Again, since \((4)^2 = 16\), the denominator becomes:
\[16 \times 2^{(n+2)} - 2 \times 2^{(n+2)}\]
\[= (16 - 2) \times 2^{(n+2)}\]
\[= 14 \times 2^{(n+2)}\]
Step 3: Combine and Simplify
Now let's plug the simplified numerator and denominator back into the expression:
\[\frac{16 \times 2^{(n+1)} - 4 \times 2^n}{14 \times 2^{(n+2)}}\]
Factor out \(2^n\) from the numerator:
\[= \frac{2^n (16 \times 2 - 4)}{14 \times 2^{(n+2)}}\]
\[= \frac{2^n (32 - 4)}{14 \times 2^{(n+2)}}\]
\[= \frac{2^n (28)}{14 \times 2^{(n+2)}}\]
Factor out \(2^{n+2}\) from the denominator:
\[= \frac{28 \times 2^n}{14 \times 2^{(n+2)}}\]
Since \(2^{(n+2)} = 4 \times 2^n\):
\[= \frac{28 \times 2^n}{14 \times 4 \times 2^n}\]
Cancel \(2^n\):
\[= \frac{28}{14 \times 4}\]
Simplify:
\[= \frac{28}{56}\]
Final Answer:
\[= \frac{1}{2}\]
Therefore, the value is 0.5.

Answer 2

We can check the simplified result by plugging in an actual value of n, since the expression should reduce to the same number for any n.

1. Step 1: Take \( n = 0 \). The numerator becomes \( 16 \times 2^1 - 4 \times 2^0 = 32 - 4 = 28 \).
2. Step 2: The denominator becomes \( 16 \times 2^2 - 2 \times 2^2 = 64 - 8 = 56 \).
3. Step 3: Dividing gives \( \frac{28}{56} = 0.5 \).
4. Step 4: Trying \( n = 1 \) as a check: numerator \( = 16 \times 4 - 4 \times 2 = 64 - 8 = 56 \), denominator \( = 16 \times 8 - 2 \times 8 = 128 - 16 = 112 \), and \( \frac{56}{112} = 0.5 \) again.

Both values of n give the same result, confirming the expression always simplifies to 0.5.

Therefore, the correct answer is 0.5.