Evaluate the sum: sum frac{1}{1 + x^{a - b} + | vedclass

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(A) Let the given expression be $S = \sum \frac{1}{1 + x^{a - b} + x^{a - c}}$.Multiplying the numerator and denominator of each term by $x^a$,we get:

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(A) Let the given expression be $S = \sum \frac{1}{1 + x^{a - b} + x^{a - c}}$.Multiplying the numerator and denominator of each term by $x^a$,we get:$S = \sum \frac{x^a}{x^a + x^b + x^c}$.Wait,the standard form for this cyclic sum is $\sum \frac{1}{1 + x^{b-a} + x^{c-a}} = \sum \frac{x^a}{x^a + x^b + x^c} = 1$ if $a+b+c=0$ or similar constraints.Given the expression $\sum \frac{1}{1 + x^{a-b} + x^{a-c}} = \sum \frac{x^b x^c}{x^b x^c + x^a x^c + x^a x^b} = \sum \frac{x^{b+c}}{x^{b+c} + x^{a+c} + x^{a+b}}$.Since the denominator $x^{a+b} + x^{b+c} + x^{c+a}$ is common for all terms in the cyclic sum:$S = \frac{1}{x^{a+b} + x^{b+c} + x^{c+a}} \sum (x^{b+c} + x^{a+c} + x^{a+b}) = 1$.

Evaluate the sum: $\sum \frac{1}{1 + x^{a - b} + x^{a - c}}$

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