JEE Mains · Maths · STD 11 - 8. sequence and series
If the sum \(\frac{3}{1^2} + \frac{5}{{{1^2} + {2^2}}} + \frac{7}{{{1^2} + {2^2} + {3^2}}} + ...... + \) up to \(20\) terms is equal to \(\frac{k}{{21}}\), then \(k\) is equal to
- A \(120\)
- B \(180\)
- C \(240\)
- D \(60\)
Answer & Solution
Correct Answer
(A) \(120\)
Step-by-step Solution
Detailed explanation
\({n^{th}}\) term of given series is \(\frac{{2n + 1}}{{\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}}} = \frac{6}{{n\left( {n + 1} \right)}}\) Let \({n^{th}}\) term, \({a_n} = 6\left[ {\frac{1}{n} - \frac{1}{{n + 1}}} \right]\) Sum of \(20\) terms,…
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
- The mean and standard deviation of 100 observations are 40 and 5.1 , respectively, By mistake one observation is taken as 50 instead of 40. If the correct mean and the correct standard deviation are \(\mu\) and \(\sigma\) respectively, then \(10(\mu+\sigma)\) is equal toJEE Mains 2025 Medium
- If \({\sum\limits_{i = 1}^{20} {\left( {\frac{{{}^{20}{C_{i - 1}}}}{{{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3}\, = \frac{k}{{21}}\), then \(k\) equalsJEE Mains 2019 Hard
- Let \(T\) be the tangent to the ellipse \(E: x^{2}+4 y^{2}=5\) at the point \(P(1,1)\). If the area of the region bounded by the tangent \(T\), ellipse \(E\), lines \(x=1\) and \(x=\sqrt{5}\) is \(\alpha \sqrt{5}+\beta+\gamma \cos ^{-1}\left(\frac{1}{\sqrt{5}}\right)\), then \(|\alpha+\beta+\gamma|\) is equal to \(....\)JEE Mains 2021 Hard
- Bag \(B_1\) contains 6 white and 4 blue balls, Bag \(B_2\) contains 4 white and 6 blue balls, and Bag \(B_3\) contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag \(B_2\), is :JEE Mains 2025 Easy
- Let \(A=\{0,1,2,3,4,5,6,7\} .\) Then the number of bijective functions \(f: A \rightarrow A\)such that \(f(1)+f(2)=3-f(3)\) is equal to \(.....\)JEE Mains 2021 Hard
- Let the set \(C=\left\{(x, y) \mid x^2-2^y=2023, x, y \in \mathbb{N}\right\}\). Then \(\sum_{(x, y) \in C}(x+y)\) is equal toJEE Mains 2024 Hard
More PYQs from JEE Mains
- The two adjacent sides of a cyclic quadrilateral are \(2\) and \(5\) and the angle between them is \(60^o\). If the area of the quadrilateral is \(4\sqrt 3 \) , then the perimeter of the quadrilateral isJEE Mains 2017 Hard
- Let \(C\) be a circle having centre in the first quadrant and touching the \(x\)-axis at a distance of \(3\) units from the origin. If the circle \(C\) has an intercept of length \(6\sqrt{3}\) on \(y\)-axis, then the length of the chord of the circle \(C\) on the line \(x - y = 3\) is :JEE Mains 2026 Medium
- Let \(S=\left\{n \in N \mid\left(\begin{array}{ll}0 & i \\ 1 & 0\end{array}\right)^{n}\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \forall a, b, c, d \in R\right\}\), where \(i=\sqrt{-1} .\) Then the number of \(2 -\) digit numbers in the set \(\mathrm{S}\) is \(......\)JEE Mains 2021 Medium
- Let \(\hat{u}\) and \(\hat{v}\) be unit vectors inclined at an acute angle such that \(|\hat{u}\times\hat{v}|=\dfrac{\sqrt{3}}{2}\). If \(\vec{A}=\lambda\hat{u}+\hat{v}+(\hat{u}\times\hat{v})\), then \(\lambda\) is equal to:JEE Mains 2026 Hard
- Let \( f(\alpha) \) denote the area of the region in the first quadrant bounded by \( x=0, x=1, y^{2}=x \) and \( y=|\alpha x-5|-|1-\alpha x|+\alpha x^{2}. \) Then \( (f(0)+f(1)) \) is equal toJEE Mains 2026 Hard
- If the solution curve, of the differential equation \(\frac{d y}{d x}=\frac{x+y-2}{x-y}\) passing through the point \((2,1)\) is \(\tan ^{-1}\left(\frac{y-1}{x-1}\right)-\frac{1}{\beta} \log _e\left(\alpha+\left(\frac{y-1}{x-1}\right)^2\right)=\log _e|x-1|\), then \(5 \beta+\alpha\) is equal toJEE Mains 2024 Hard