If ` P_(r)` is the coefficient of ` x^®` in the expansion of
`(1 + x)^(2) (1 + (x)/(2))^(2) (1 + (x)/(2^(2)))^(2) (1 + (x)/(2^(3)))^(2)...` prove that
`P_(r) = (2^(2))/((2^(r) -1))(P_(r-1) + P_(r-2) )and P_(4) = (1072)/(315)` .

Explanation

To solve the problem, we need to prove that \( P_r = \frac{4}{2^r - 1}(P_{r-1} + P_{r-2}) \) and find \( P_4 = \frac{1072}{315} \).

### Step 1: Understand the given expression
The expression we are dealing with is:
\[ (1 + x)^2 (1 + \frac{x}{2})^2 (1 + \frac{x}{2^2})^2 (1 + \frac{x}{2^3})^2 \ldots \]
This can be interpreted as a product of binomials, where each term contributes to the expansion.
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