Let a _ n be a sequence such that a_1+a_2+ +a_n=n^2+3 n (n+1)(n+2… - Vidyadip

Question

Let $\left\langle a _{ n }\right\rangle$ be a sequence such that $a_1+a_2+\ldots+a_n=\frac{n^2+3 n}{(n+1)(n+2)}$. If $28 \sum \limits_{ k =1}^{10} \frac{1}{ a _{ k }}= p _1 p _2 p _3 \ldots p _{ m }$, where $p _1, p _2, \ldots . pm$ are the first $m$ prime numbers, then $m$ is equal to

✓

Answer

b
$a_n=S_n-S_{n-1}=\frac{ n ^2+3 n }{( n +1)( n +2)}-\frac{( n -1)( n +2)}{ n ( n +1)}$

$\Rightarrow a _{ n }=\frac{4}{ n ( n +1)( n +2)}$

$\Rightarrow 28 \sum \limits_{ k =1}^{10} \frac{1}{ a _{ k }}=28 \sum \limits_{ k =1}^{10} \frac{ k ( k +1)( k +2)}{4}$

$=\frac{7}{4} \sum \limits_{ k =1}^{10}( k ( k +1)( k +2)( k +3)-( k -1) k ( k +1)( k +2))$

$=\frac{7}{4} \cdot 10 \cdot 11 \cdot 12 \cdot 13=2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13$

So $m =6$

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 total number of words $($with or without meaning$)$ that can be formed out of the letters of the word $\text{‘DISTRIBUTION’}$ taken four at a time, is equal to $..........$

Number of integral values of $a$ for which both roots of quadratic equation $x^2 -(2a + 3)x + a^2 + 3a = 0$ lies in interval $(0,4)$ is-

Let $f(\mathrm{x})=\left|2 \mathrm{x}^2+5\right| \mathrm{x}|-3|, \mathrm{x} \in \mathrm{R}$. If $\mathrm{m}$ and $\mathrm{n}$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m+n$ is equal to :

Let $A_{1}=\left\{(x, y):|x| \leq y^{2},|x|+2 y \leq 8\right\}$ and $A_{2}=\{(x, y):|x|+|y| \leq k\}$. If $27$ (Area $\left.A _{1}\right)=5$ (Area $A _{2}$ ), then $k$ is equal to

For $n \in N$, if $\cot ^{-1} 3+\cot ^{-1} 4+\cot ^{-1} 5+\cot ^1 n =\frac{\pi}{4}$, then $n$ is equal to$.......$

The sum to infinite term of the series $1 + \frac{2}{3} + \frac{6}{{{3^2}}} + \frac{{10}}{{{3^3}}} + \frac{{14}}{{{3^4}}} + \ldots \;$ is

A circle has centre $C$ on axes of parabola & it touches the parabola at point $P$. $CP$ makes an angle of $120^o$ with axis of parabola. If radius of circle is $2$, then latus rectum of parabola is-

The length of tangent from the point $(5, 1)$ to the circle ${x^2} + {y^2} + 6x - 4y - 3 = 0$, is

Let $a, b, c, d, e$ be natural numbers in an arithmetic progression such that $a+b+c+d+e$ is the cube of an integer and $b+c+d$ is square of an integer. The least possible value of the number of digits of $c$ is

Let $A=\left[\begin{array}{lll}2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{array}\right], B=\left[B_1, B_2, B_3\right]$, where $B_1, \mathrm{B}_2, \mathrm{~B}_3$ are column matrices, and $\mathrm{AB}_1=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right], \mathrm{AB}_2=\left[\begin{array}{l}2 \\ 3 \\ 0\end{array}\right], \mathrm{AB}_3=\left[\begin{array}{l}3 \\ 2 \\ 1\end{array}\right]$ If $\alpha=|B|$ and $\beta$ is the sum of all the diagonal elements of $B$, then $\alpha^3+\beta^3$ is equal to