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8. Sequences and Series
hard
अनुक्रम $\frac{1}{3}, \frac{5}{9}, \frac{19}{27}, \frac{65}{81}, \ldots \ldots$ के प्रथम $100$ पदों के योगफल से छोटा या बराबर महत्तम पूर्णांक होगा
A
$99$
B
$98$
C
$89$
D
$88$
(JEE MAIN-2022)
Solution
$\frac{1}{3}+\frac{5}{9}+\frac{19}{27}+\frac{65}{81}+\ldots$
$\left(1-\frac{2}{3}\right)+\left(1-\frac{4}{9}\right)+\left(1-\frac{8}{27}\right)+\left(1-\frac{16}{81}\right) \ldots .100 \text { terms }$
$100-\left[\frac{2}{3}+\left(\frac{2}{3}\right)^{2}+\ldots\right]$
$100-\frac{2}{3}\left(1-\left(\frac{2}{3}\right)^{100}\right)$
$100-2\left(1-\left(\frac{2}{3}\right)^{100}\right)$
$S =98+2\left(\frac{2}{3}\right)^{100}$
$\Rightarrow[ S ]=98$
Standard 11
Mathematics
