Let \( \alpha, \beta, \gamma \) and \( \delta \) be the coefficients of \( x^7, x^5, x^3, x \) respectively in the expansion of \( (x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, \, x > 1 \). If \( \alpha u + \beta v = 18 \), \( \gamma u + \delta v = 20 \), then \( u + v \) equals:
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For problems involving binomial expansions, it's crucial to recall the binomial theorem, which states that:
\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k.
\]
For such expansions, focus on finding the relevant coefficients and use the given relationships between the coefficients to form equations. This will help in solving for the unknowns \( u \) and \( v \) in this case.
The expansion of \( (x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5 \) is provided. This expression is a binomial expansion.The expression can be divided into two components:\[(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5\]Applying the binomial theorem, each term is expanded. We are focused on the coefficients of \( x^7, x^5, x^3, x \).The relevant binomial expansions yield the coefficients \( \alpha, \beta, \gamma, \delta \).With these coefficients determined, the equations \( \alpha u + \beta v = 18 \) and \( \gamma u + \delta v = 20 \) constitute a system of equations.Solving this system for \( u \) and \( v \) involves substituting the values of \( \alpha, \beta, \gamma, \delta \).Upon solving the system, the result obtained is:\[u + v = 5.\]
