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  4. If T= displaystyle∑r=118 ((-1)r-1 ⋅ r ⋅ 18 Cr/(r+1)) then ((1/T)-10) is equal to

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Q. If $T=\displaystyle\sum_{r=1}^{18} \frac{(-1)^{r-1} \cdot r \cdot{ }^{18} C_{r}}{(r+1)}$ then $\left(\frac{1}{T}-10\right)$ is equal to

Binomial Theorem

Solution:

$\frac{r}{(r+1)} \cdot{ }^{18} C_{r}=\left(1-\frac{1}{r+1}\right) \cdot{ }^{18} C_{r}$
$={ }^{18} C_{r}-\frac{\lfloor 18}{(r+1)\lfloor r\lfloor 18-r}$
$={ }^{18} C_{r}-\frac{1}{19}{ }^{19} C_{r+1}$
$\therefore T=\displaystyle\sum_{r=1}^{18}(-1)^{r-1} \cdot \frac{r}{(r+1)} \cdot{ }^{18} C_{r}$
$={ }^{18} C_{1}-{ }^{18} C_{2}+{ }^{18} C_{3}-\ldots$
$={ }^{18} C_{0}-\frac{1}{19}\left(-{ }^{19} C_{0}+{ }^{19} C_{1}\right)$
$=1+\frac{1}{19}-1=\frac{1}{19}$
$\Rightarrow \frac{1}{T}=19$