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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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:

Vedclass · Accepted Answer

$1$

Vedclass · Answer

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

Similar Questions

If $\cosh \beta = \sec \alpha \cos \theta$ and $\sinh \beta = \operatorname{cosec} \alpha \sin \theta$,then $\sinh^2 \beta =$

The value of $\cos 52^\circ + \cos 68^\circ + \cos 172^\circ$ is

The value of $\cos ^2 76^{\circ}+\cos ^2 16^{\circ}-\cos 76^{\circ} \cos 16^{\circ}$ is equal to

If $3 \sin \alpha = 5 \sin \beta$,then $\tan \left(\frac{\alpha + \beta}{2}\right) \div \tan \left(\frac{\alpha - \beta}{2}\right) = $

If $\sin 2\theta + \sin 2\phi = \frac{1}{2}$ and $\cos 2\theta + \cos 2\phi = \frac{3}{2}$,then $\cos^2(\theta - \phi) =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module