Download App

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

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 8. sequence and series4 Marks
Question
Let $[\alpha]$ denote the greatest integer $\leq \alpha$. Then $[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots .+[\sqrt{120}]$ is equal to.
✓

Answer

b
$[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots \ldots .[\sqrt{120}]$

$\Rightarrow 1+1+1+2+2+2+2+2+3+3+\ldots \ldots .+$

$3=7 \text { times }$

$+4+4+\ldots \ldots .+4=9 \text { times }+\ldots \ldots 10+10+$

$\ldots \ldots+10=21 \text { times }$

$\Rightarrow \sum_{r=1}^{10}(2 r+1) . r$

$\Rightarrow 2 \sum_{r=1}^{10} r^2+\sum_{r=1}^{10} r$

$\Rightarrow 2 \times \frac{10 \times 11 \times 21}{6}+\frac{10 \times 11}{2}$

$\Rightarrow 770+55$

$\Rightarrow 825$

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

Let $a _1, a _2, a _3, \ldots$ be a $G.P.$ of increasing positive numbers. Let the sum of its $6^{\text {th }}$ and $8^{\text {th }}$ terms be $2$ and the product of its $3^{\text {rd }}$ and $5^{\text {th }}$ terms be $\frac{1}{9}$. Then $6\left( a _2+\right.$ $\left.a_4\right)\left(a_4+a_6\right)$ is equal to
→
$\mathop {\lim }\limits_{x \to {2^ + }}  \frac{{1 - \cos \{ {x^2} + 2x\} }}{{\ln {{(x - 1)}^{(x - 2)}}}}$ is equal to

(where $\{.\}$ denotes fractional part function) 

→
The lines $\frac{x-2}{2}=\frac{y}{-2}=\frac{z-7}{16}$ and $\frac{x+3}{4}=\frac{y+2}{3}=\frac{z+2}{1}$ intersect at the point $P$. If the distance of $P$ from the line $\frac{x+1}{2}=\frac{y-1}{3}=\frac{z-1}{1}$ is $l$, then $14 l^2$ is equal to.................
→
If the edge of a cube increases at the rate of  $60\, cm$ per second, at what rate the volume is increasing when the edge is $ 90\, cm$ .......... $cu\, cm \,per\, sec$
→
If $^n{C_r} = 84,{\;^n}{C_{r - 1}} = 36$ and $^n{C_{r + 1}} = 126$, then $n$ equals
→
If ${}^n{P_4}\;:\;{}^n{P_5} = 1:2$, then $n = $
→
If $f (x) = \frac{{{{\log }_{\sin |x|}}{{\cos }^3}x}}{{{{\log }_{\sin |3x|}}{{\cos }^3}\left( {\frac{x}{2}} \right)}}for |x| <\frac{\pi }{3} x \ne 0= 4$ for $x = 0$then, the number of points of discontinuity of f in $\left( { - \frac{\pi }{3},\,\frac{\pi }{3}} \right)$ is
→
Let the line $L : \sqrt{2} x + y =\alpha$ pass through the point of the intersection $P ($in the first quadrant$)$ of the circle $x ^2+ y ^2=3$ and the parabola $x ^2=2 y$. Let the line $L$ touch two circles $C_1$ and $C_2$ of equal radius $2 \sqrt{3}$. If the centres $Q _1$ and $Q _2$ of the circles $C _1$ and $C _2$ lie on the $y-$ axis, then the square of the area of the triangle $PQ _1 Q _2$ is equal to $.....$
→
The number of ways in which any four letters can be selected from the word ‘$CORGOO$’ is
→
Let $A (-1,1), B (3,4)$ and $C (2,0)$ be given three points. $A$ line $y = mx , m > 0$, intersects lines $AC$ and $BC$ at point $P$ and $Q$ respectively. Let $A _{1}$ and $A _{2}$ be the areas of $\Delta ABC$ and $\Delta PQC$ respectively, such that $A _{1}=3 A _{2}$, then the value of $m$ is equal to:
→