If $S$ be the sum of coefficients in the expansion of ${(\alpha x + \beta y - \gamma\mathcal{z)}}^{n}$, where $(\alpha,\beta,\gamma 0)$, then the value of $\lim_{n \rightarrow \infty}\frac{S}{{\{ S^{\frac{1}{n}} + 1\}}^{n}}$ is

Question

If $S$ be the sum of coefficients in the expansion of ${(\alpha x + \beta y - \gamma\mathcal{z)}}^{n}$, where $(\alpha,\beta,\gamma > 0)$, then the value of $\lim_{n \rightarrow \infty}\frac{S}{{\{ S^{\frac{1}{n}} + 1\}}^{n}}$ is

Dr. Abubacker · Accepted Answer

(d)
$S = {(\alpha \bullet 1 + \beta \bullet 1 + \gamma \bullet 1)}^{n} = {(\alpha + \beta - \gamma)}^{n}$
$	herefore\ \lim_{n ightarrow \infty}{\frac{s}{\left\{ S^{\frac{1}{n}} + 1 ight\}^{n}} = \lim_{n ightarrow \infty}{\left( \frac{\alpha + \beta - \gamma}{\alpha + \beta - \gamma + 1} ight)^{n} = 0}}$
$(\because\ \alpha + \beta - \gamma + 1 > \alpha + \beta - \gamma)$

Q. If $S$ be the sum of coefficients in the expansion of ${(\alpha x + \beta y - \gamma\mathcal{z)}}^{n}$, where $(\alpha,\beta,\gamma > 0)$, then the value of $\lim_{n \rightarrow \infty}\frac{S}{{\{ S^{\frac{1}{n}} + 1\}}^{n}}$ is

Explanation

Reviewed by Dr. Abubacker

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