MCQ
If $x = \sum\limits_{n = 0}^\infty {{a^n}} ,\;y = \sum\limits_{n = 0}^\infty {{b^n},\;z = \sum\limits_{n = 0}^\infty {{{(ab)}^n}} } $, where $a,\;b < 1$, then
- A$xyz = x + y + z$
- ✓$xz + yz = xy + z$
- C$xy + yz = xz + y$
- D$xy + xz = yz + x$
$\Rightarrow a = \frac{{x - 1}}{x}$
$y = \sum\limits_{n = 0}^\infty {{b^n}} = \frac{1}{{1 - b}}$
$ \Rightarrow $ $b = \frac{{y - 1}}{y}$
$z = \sum\limits_{n = 0}^\infty {{a^n}{b^n} = \frac{1}{{1 - ab}} \Rightarrow ab = \frac{{z - 1}}{z}} $
$\therefore $ $\frac{{x - 1}}{x}.\frac{{y - 1}}{y} = \frac{{z - 1}}{z}$
$ \Rightarrow $ $xy + z = zx + yz$.
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$\operatorname{det}\left[\begin{array}{cc}\sum_{k=0}^n k & \sum_{k=0}^n{ }^n C_k k^2 \\ \sum_{k=0}^n{ }^n C_k k & \sum_{k=0}^n{ }^n C_k 3^k\end{array}\right]=0$, holds for some positive integer $n$. Then $\sum_{k=0}^n \frac{{ }^n C_k}{k+1}$ equals