MCQ
If $y = x - {x^2} + {x^3} - {x^4} + ......\infty $, then value of $x$ will be
- A$y + \frac{1}{y}$
- B$\frac{y}{{1 + y}}$
- C$y - \frac{1}{y}$
- ✓$\frac{y}{{1 - y}}$
then $xy = {x^2} - {x^3} + {x^4} - ......\infty $
Adding, $y + xy = x + 0 + 0...... + 0$
$ \Rightarrow $$x - xy = y $
$\Rightarrow x(1 - y) = y$
$\Rightarrow x = \frac{y}{{1 - y}}$.
Aliter : $y = \frac{x}{{1 - ( - x)}} $
$\Rightarrow y = \frac{x}{{1 + x}}$
$ \Rightarrow $$y + yx = x$
$\Rightarrow x = \frac{y}{{1 - y}}$.
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$S _1=\{( i , j , k ): i , j , k \in\{1,2, \ldots, 10\}\}$
$S _2=\{( i , j ): 1 \leq i < j +2 \leq 10, i , j \in\{1,2, \ldots, 10\}\},$
$S _3=\{( i , j , k , l): 1 \leq i < j < k < l, i , j , k , l \in\{1,2, \ldots ., 10\}\}$
$S _4=\{( i , j , k , l): i , j , k$ and $l$ are distinct elements in $\{1,2, \ldots, 10\}\}$
and If the total number of elements in the set $S _t$ is $n _z, r =1,2,3,4$, then which of the following statements is (are) TRUE?
$(A)$ $n _1=1000$ $(B)$ $n _2=44$ $(C)$ $n _3=220$ $(D)$ $\frac{ n _4}{12}=420$
$7 \times 8,10 \times 10,13 \times 12,16 \times 14, \ldots .$ is ....... .