JEE Mains · Maths · STD 11 - 8. sequence and series
Let \(S_n\) be the sum to n-terms of an arithmetic progression \(3,7,11, \ldots\) If \(40<\left(\frac{6}{\mathrm{n}(\mathrm{n}+1)} \sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{S}_{\mathrm{k}}\right)<42\), then \(\mathrm{n}\) equals
- A \(9\)
- B \(8\)
- C \(10\)
- D \(7\)
Answer & Solution
Correct Answer
(A) \(9\)
Step-by-step Solution
Detailed explanation
\(S_n= 3+7+11+............ n \) terams \({n}{2}(6+(n-1) 4)=3 n+2 n^2-2 n \) \( =2 n^2+n \) \( \sum_{k=1}^n S_k=2 \sum_{k=1}^n K^2+\sum_{k=1}^n K \) \( =2 \cdot \frac{n(n+1)(2 n+1)}{6}+\frac{n(n+1)}{2} \) \(=n(n+1)\left[\frac{2 n+1}{3}+\frac{1}{2}\right]\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If the Rolle's theorem holds for the function \(f(x) = 2x^3 + ax^2 + bx\) in the interval \([-1, 1 ]\) for the point \(c = \frac{1}{2}\) , then the value of \(2a + b\) isJEE Mains 2015 Hard
- The number of distinct real roots of \(\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0\) in the interval \(-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}\) isJEE Mains 2021 Hard
- Let the point \(P\) be the vertex of the parabola \(y = x^2 - 6x + 12\). If a line passing through the point \(P\) intersects the circle \(x^2 + y^2 - 2x - 4y + 3 = 0\) at the points \(R\) and \(S\), then the maximum value of \((PR + PS)^2\) is :JEE Mains 2026 Medium
- If \(\int {{e^{\sec \,x}}\,\left( {\sec \,x + \tan \,x\,f\left( x \right) + \left( {\sec \,x\,\tan \,x + {{\sec }^2}\,x} \right)} \right)dx = {e^{\sec \,x\,}}\,f\left( x \right)} + C\) , then a possible choice of \(f\left( x \right)\) isJEE Mains 2019 Hard
- Let the eccentricity of the hyperbola \(H : \frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1\) be \(\sqrt{\frac{5}{2}}\) and length of its latus rectum be \(6 \sqrt{2}\), If \(y =2 x + c\) is a tangent to the hyperbola \(H\), then the value of \(c ^{2}\) is equal toJEE Mains 2022 Hard
- For \(n \in N\), let \(S _{ n }=\left\{ z \in C :| z -3+2 i |=\frac{ n }{4}\right\}\) and \(T _{ n }=\left\{ z \in C :| z -2+3 i |=\frac{1}{ n }\right\}\) Then the number of elements in the set \(\left\{ n \in N : S _{ a } \cap T _{ n }=\phi\right\}\) is.JEE Mains 2022 Hard
More PYQs from JEE Mains
- Number of functions \(f:\{1,2, \ldots, 100\} \rightarrow\{0,1\}\), that assign 1 to exactly one of the positive integers less than or equal to 98 , is equal to ________.JEE Mains 2025 Medium
- The integral \(\int {\frac{{xdx}}{{2 - {x^2} + \sqrt {2 - {x^2}} }}} \) equalsJEE Mains 2013 Hard
- Let \(f: R \rightarrow R\) be defined as \(f(x)=\left[\begin{array}{ll}{\left[e^{x}\right],} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,x<0 \\ a e^{x}+[x-1], \,\,\,\,\,\,\,\,\,0 \leq x<1 \\ b+[\sin (\pi x)], \,\,\,\,\,\,\,\,\,\,\,\,1 \leq x<2 \\ {\left[e^{-x}\right]-c,} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,x \geq 2\end{array}\right.\) where a,b,c \(\in R\) and \([t]\) denotes greatest integer less than or equal to \(t.\) Then, which of the following statements is true \(?\)JEE Mains 2022 Hard
- Let \(r_{1}\) and \(r_{2}\) be the radii of the largest and smallest circles, respectively, which pass through the point \((-4,1)\) and having their centres on the circumference of the circle \(x^{2}+y^{2}+2 x+4 y-4= 0.\) If \(\frac{r_{1}}{r_{2}}=a+b \sqrt{2}\), then \(a+b\) is equal to:JEE Mains 2021 Hard
- 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 toJEE Mains 2022 Hard
- Let \([ x ]\) denote greatest integer less than or equal to \(x .\) If for \(n \in N ,\left(1-x+x^{3}\right)^{n}=\sum_{j=0}^{3 n} a_{j} x^{j}\), then \(\sum_{j=0}^{\left[\frac{3 n}{2}\right]} a_{2 j}+4 \sum_{j=0}^{\left[\frac{3 n-1}{2}\right]} a_{2 j+1}\) is equal toJEE Mains 2021 Hard