For a real number $$ x $$, if $$ \frac{1}{2}, \frac{\log_4(x^2 - 9)}{\log_4 4}, ext{ and } \frac{\log_4\left(\frac{x^2 + 17}{2} ight)}{\log_4 4} $$ are in an arithmetic progression, then the common difference is:

Question

For a real number $$ x $$, if $$ \frac{1}{2}, \frac{\log_4(x^2 - 9)}{\log_4 4}, 	ext{ and } \frac{\log_4\left(\frac{x^2 + 17}{2}ight)}{\log_4 4} $$ are in an arithmetic progression, then the common difference is:

Accepted Answer

The given terms are in arithmetic progression (AP). Let the terms be $$ A = \frac{1}{2}, B = \frac{\log_4(x^2 - 9)}{\log_4 4}, C = \frac{\log_4\left(\frac{x^2 + 17}{2}ight)}{\log_4 4} $$.

From the property of AP, the difference between consecutive terms is constant:  
$$ B - A = C - B. $$

This implies that the terms $$ x^2 - 9, x^2 + 17/2 $$, and their logarithms form a geometric progression (GP). Solving the resulting equations step-by-step:

1. Substitute $$ y = x^2 $$ to simplify calculations.
2. Solve the quadratic equation derived from GP conditions:  
   $$ y^2 - 20y + 64 = 0. $$
3. The solutions are $$ y = 16 	ext{ or } y = 4. $$
   Since the argument of logarithms cannot be negative, $$ x^2 = 16, x = 4. $$
4. Compute the common ratio of GP and hence determine the common difference of AP:  
   $$ d = B - A = C - B = \log_4\left(\frac{7}{2}ight). $$

Thus, the common difference is $$ \log_4\left(\frac{7}{2}ight) $$.

Answer

Answer

Answer