On simplification the product (x + y)(x² + y²)(x⁴ + y⁴)(x⁸ + y⁸) ... h

On simplification the product (x + y)(x² + y²)(x⁴ + y⁴)(x⁸ + y⁸) … how many such terms are there which will have only single x and rest ys?

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UPSC CAPF – 2018

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The product is (x + y)(x² + y²)(x⁴ + y⁴)(x⁸ + y⁸)… Let’s consider a product of n terms: P_n = (x^2^0 + y^2^0)(x^2^1 + y^2^1)…(x^2^(n-1) + y^2^(n-1)). When expanding this product, each term is formed by choosing either the first term (x raised to a power of 2) or the second term (y raised to the same power of 2) from each bracket. For example, from the i-th bracket (x^2^i + y^2^i), we pick either x^2^i or y^2^i. A term in the expansion is the product of one chosen term from each of the n brackets. We are looking for terms with a “single x”, which means the power of x in the term must be 1 (x¹). Let’s say we pick x^2^i from one bracket and y^2^j from all other brackets (j ≠ i). The resulting term is (x^2^i) * (product of y^2^j for all j ≠ i). The power of x in this term is 2^i. For this power to be 1 (x¹), we must have 2^i = 1. This occurs only when i = 0. So, the only way to get a term with x¹ is to choose x^2^0 = x from the first bracket (x+y) and y^2^j from all subsequent brackets (j = 1, 2, …, n-1). The resulting term is x * y² * y⁴ * y⁸ * … * y^(2^(n-1)) = x * y^(2¹ + 2² + … + 2^(n-1)). The power of y is the sum of a geometric series: 2(2^(n-1) – 1)/(2-1) = 2^n – 2. The term is x¹y^(2^n – 2). This is the only term in the expansion that has x raised to the power of 1. Any other combination will result in x raised to a power other than 1 (e.g., choosing x from multiple brackets, or choosing x^2^i where i > 0 results in a power 2^i > 1). Thus, there is only exactly one such term. The number of such terms is 1.

In the expansion of a product of factors of the form (x^a + y^a), terms are formed by selecting one element from each factor and multiplying them. To find terms with a specific power of x, analyze the possible combinations of selections from each factor.

This type of product (x+y)(x²+y²)(x⁴+y⁴)… is related to the identity (x-y)(x+y) = x²-y², (x²-y²)(x²+y²) = x⁴-y⁴, and generally (x^2^n – y^2^n)(x^2^n + y^2^n) = x^2^(n+1) – y^2^(n+1). If the product were multiplied by (x-y), it would telescope nicely to x^(2^n) – y^(2^n). However, the question asks about the terms in the expansion of the product itself.

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