Download App

STD 11 - 8. sequence and series — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 8. sequence and series4 Marks
MCQ
 If $\left(\frac{1}{\alpha+1}+\frac{1}{\alpha+2}+\ldots+\frac{1}{\alpha+1012}\right) $ $ -\left(\frac{1}{2 \cdot 1}+\frac{1}{4 \cdot 3}+\frac{1}{6 \cdot 5}+\ldots+\frac{1}{2024 \cdot 2023}\right) $

$ =\frac{1}{2024}, $ then $\alpha$ is equal to-

  • A
    $1367$
  • B
    $1058$
  • C
    $1056$
  • ✓
    $1011$
✓

Answer

Correct option: D.
$1011$
d
$ \left(\frac{1}{\alpha+1}+\frac{1}{\alpha+2}+\ldots+\frac{1}{\alpha+2012}\right) $ - $ \left\{\left(\frac{1}{1}-\frac{1}{2}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\ldots+\left(\frac{1}{2023}-\frac{1}{2024}\right)\right\}=\frac{1}{2024} $

$\Rightarrow $ $ \left(\frac{1}{\alpha+1}+\frac{1}{\alpha+2}+\ldots+\frac{1}{\alpha+2012}\right) $ $ -\left\{\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)+\ldots+\frac{1}{2023}\right. $ $ \left.-\frac{1}{2024}-2\left(\frac{1}{2}+\frac{1}{4}+\ldots+\frac{1}{2022}\right)\right\}=\frac{1}{2024} $

$\Rightarrow $ $ \left(\frac{1}{\alpha+1}+\frac{1}{\alpha+2}+\ldots+\frac{1}{\alpha+2012}\right) $ $ -\left(\frac{1}{1}+\frac{1}{2}+\ldots+\frac{1}{2023}\right) $ $ \frac{1}{2024}+\left(\frac{1}{1}+\frac{1}{2}+\ldots+\frac{1}{1011}\right)=\frac{1}{2024} $

$\Rightarrow $ $ \frac{1}{\alpha+1}+\frac{1}{\alpha+2}+\ldots+\frac{1}{\alpha+2012} $ = $ \frac{1}{1012}+\frac{1}{1013}+\ldots+\frac{1}{2023} $

$\Rightarrow $ $ \alpha=1011$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The middle term in the expansion of ${\left( {x + \frac{1}{{2x}}} \right)^{2n}}$, is
→
Equation of the perpendicular bisector of the line segment joining the points $(7, 4)$ and $(-1, -2)$, is
→
Let $\alpha $ and $\beta $ be integers satisfying $0 < \beta < \alpha $ .Let $P\left( {\alpha ,\beta } \right),Q$ be the reflection of $P$ in the line $y = x, R$ be the reflection of $Q$ in the $y-$ axis, $S$ be the reflection of $R$ in the $x-$ axis and $T$ be the reflection of $S$ in the $y-$ axis. If the area of convex pentagon $PQRST$ is $187\ sq. units$ , then value of $\alpha  + {\beta ^2}$ is
→
Let $\alpha=1^2+4^2+8^2+13^2+19^2+26^2+\ldots \ldots$ upto 10 terms and $\beta=\sum_{n=1}^{10} n^4$. If $4 \alpha-\beta=55 k+40$, then $\mathrm{k}$ is equal to____________.
→
If coordinates of the points $A$ and $ B$ are $(2, 4)$ and $(4, 2)$ respectively and point M is such that $A-M-B$ also $AB =\, 3\, AM$, then the coordinates of $ M$ are
→
In a box containing $100$ eggs, $10$ eggs are rotten. The probability that out of a sample of $5$ eggs none is rotten if the sampling is with replacement is
→
Let $A = \left\{ {\theta :\,2{{\cos }^2}\theta  + \sin \theta  \le 2} \right\}$ and $B = \left\{ {\theta :\,\frac{\pi }{2} \le \theta  \le \frac{{3\pi }}{2}} \right\}$ , then $A \cap B$ is
→
Matrix $A = \left[ {\begin{array}{*{20}{c}}
  x&3&2 \\ 
  1&y&4 \\ 
  2&2&z 
\end{array}} \right]$, $xyz = 60$ and $8x + 4y + 3z = 20$, then $A.(Adj A)$ is equal to
→
The differential coefficient of ${\tan ^{ - 1}}\left( {{{\sqrt {1 + x} - \sqrt {1 - x} } \over {\sqrt {1 + x} + \sqrt {1 - x} }}} \right)$ is
→
$\int\limits_{ - 1}^1 {\frac{{{x^4}}}{{1 + {e^{{x^7}}}}}dx\,}= $
→