The greatest integer less than or equal to th | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

$\frac{1}{3}+\frac{5}{9}+\frac{19}{27}+\frac{65}{81}+\ldots$

$\left(1-\frac{2}{3}\right)+\left(1-\frac{4}{9}\right)+\left(1-\frac{8}{27}\right)+\le

Vedclass · Accepted Answer

$98$

Vedclass · Answer

$\frac{1}{3}+\frac{5}{9}+\frac{19}{27}+\frac{65}{81}+\ldots$

$\left(1-\frac{2}{3}ight)+\left(1-\frac{4}{9}ight)+\left(1-\frac{8}{27}ight)+\left(1-\frac{16}{81}ight) \ldots .100 	ext { terms }$

$100-\left[\frac{2}{3}+\left(\frac{2}{3}ight)^{2}+\ldotsight]$

$100-\frac{2}{3}\left(1-\left(\frac{2}{3}ight)^{100}ight)$

$100-2\left(1-\left(\frac{2}{3}ight)^{100}ight)$

$S =98+2\left(\frac{2}{3}ight)^{100}$

$\Rightarrow[ S ]=98$

The greatest integer less than or equal to the sum of first $100$ terms of the sequence $\frac{1}{3}, \frac{5}{9}, \frac{19}{27}, \frac{65}{81}, \ldots \ldots$ is equal to

Similar Questions

If the third term of a $G.P.$ is $4$ then the product of its first $5$ terms is

If $x,\;y,\;z$ are in $G.P.$ and ${a^x} = {b^y} = {c^z}$, then

The first and last terms of a $G.P.$ are $a$ and $l$ respectively; $r$ being its common ratio; then the number of terms in this $G.P.$ is

The roots of the equation

$x^5 - 40x^4 + px^3 + qx^2 + rx + s = 0$ are in $G.P.$ The sum of their reciprocals is $10$. Then the value of $\left| s \right|$ is

Let $S = N \cup\{0\}$. Define a relation $R$ from S to $R$ by: $R =\left\{(x, y): \log _e y=x \log _e\left(\frac{2}{5}\right), x \in S, y \in R \right\}$ Then, the sum of all the elements in the range of $R$ is equal to