Question

A function \(f\) satisfies the relation \( f(n^2) = f(n) + 6 \) for \(n \ge 2\) and \( f(2) = 8 \). Then the value of \( f(256) \) is

• A. 24
• B. 26
• C. 22
• D. 28
• E. 32

Hint:

Whenever you see \(f(n^2)=f(n)+k\), express the number as repeated squaring of a base value. Count how many squaring steps are needed — each step adds a constant.

Answer 1

The Correct Option is B

Concept: The given functional equation: \[ f(n^2) = f(n) + 6 \] is a recursive relation where the value of the function at a square depends on the value at its base. This type of relation is best solved by:
• Expressing the required number as repeated squaring
• Starting from the known base value
• Applying the relation step-by-step

Step 1: Analyze the structure of the number 256
We are asked to find: \[ f(256) \] Observe that: \[ 256 = 2^8 \] Now rewrite 256 in terms of repeated squaring: \[ 2 \rightarrow 2^2 = 4 \rightarrow 4^2 = 16 \rightarrow 16^2 = 256 \] So: \[ 256 = (((2)^2)^2)^2 \] This shows that we can repeatedly apply the relation starting from \(f(2)\).

Step 2: Start from the given base value
We are given: \[ f(2) = 8 \] This is the starting point for building all higher values.

Step 3: Apply the relation to compute \(f(4)\)
Since: \[ 4 = 2^2 \] Using the relation: \[ f(4) = f(2) + 6 \] Substitute: \[ f(4) = 8 + 6 = 14 \]

Step 4: Compute \(f(16)\)
Since: \[ 16 = 4^2 \] Apply relation again: \[ f(16) = f(4) + 6 \] Substitute: \[ f(16) = 14 + 6 = 20 \]

Step 5: Compute \(f(256)\)
Since: \[ 256 = 16^2 \] Apply relation: \[ f(256) = f(16) + 6 \] Substitute: \[ f(256) = 20 + 6 = 26 \]

Step 6: Verify pattern consistency
Each squaring step increases the function value by 6: \[ f(2) = 8 \] \[ f(4) = 14 \] \[ f(16) = 20 \] \[ f(256) = 26 \] We performed exactly 3 squaring steps, so: \[ 8 + 3 \times 6 = 26 \] This confirms correctness.

Step 7: Final Answer
\[ \boxed{26} \]