JEE Mains · Maths · STD 11 - 8. sequence and series
\(\frac{1}{3^{2}-1}+\frac{1}{5^{2}-1}+\frac{1}{7^{2}-1}+\ldots+\frac{1}{(201)^{2}-1}\) is equal to
- A \(\frac{101}{404}\)
- B \(\frac{25}{101}\)
- C \(\frac{101}{408}\)
- D \(\frac{99}{400}\)
Answer & Solution
Correct Answer
(B) \(\frac{25}{101}\)
Step-by-step Solution
Detailed explanation
\(T_{n}=\frac{1}{(2 n+1)^{2}-1} \frac{1}{(2 n+2) 2 n}=\frac{1}{4(n)(n+1)}\) \(=\frac{(n+1)-n}{4 n(n+1)}=\frac{1}{4}\left(\frac{1}{n}-\frac{1}{n+1}\right)\) \(S=\frac{1}{4}\left(1-\frac{1}{101}\right)=\frac{1}{4}\left(\frac{100}{101}\right)=\frac{25}{101}\)
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
- Let \(p , q \in R\) and \((1-\sqrt{3} i )^{200}=2^{199}( p + iq )\), \(i =\sqrt{-1}\) Then \(p + q + q ^2\) and \(p - q + q ^2\) are roots of the equation.JEE Mains 2023 Hard
- Let the arc \(A C\) of a circle subtend a right angle at the centre \(O\). If the point \(B\) on the arc \(A C\), divides the arc \(A C\) such that \(\frac{\text { length of } \operatorname{arc} A B}{\text { length of } \operatorname{arc} B C}=\frac{1}{5}\), and \(\overrightarrow{O C}=\alpha \overrightarrow{O A}+\beta \overrightarrow{O B}\), then \(\alpha+\sqrt{2}(\sqrt{3}-1) \beta\) is equal toJEE Mains 2025 Easy
- Let \(f(x)\) be a polynomial of degree \(3\) such that \(f(-1)=10, f(1)=-6, f(\mathrm{x})\) has a critical point at \(\mathrm{x}=-1\) and \(f^{\prime}(\mathrm{x})\) has a critical point at \(\mathrm{x}=1\) Then \(f(x)\) has a local minima at \(x=\)JEE Mains 2020 Hard
- The tangents to the curve \(y = (x -2)^2 -1\) at its points of intersection with the line \(x -y = 3\), intersect at the pointJEE Mains 2019 Hard
- Suppose A and B are the coefficients of \(30^{\text {th }}\) and \(12^{\text {th }}\) terms respectively in the binomial expansion of \((1+x)^{2 \mathrm{n}-1}\). If \(2 \mathrm{~A}=5 \mathrm{~B}\), then n is equal to :JEE Mains 2025 Medium
- If the chord joining the points \(P_{1}(x_{1},y_{1})\) and \(P_{2}(x_{2},y_{2})\) on the parabola \(y^{2}=12x\) subtends a right angle at the vertex of the parabola, then \(x_{1}x_{2}-y_{1}y_{2}\) is equal toJEE Mains 2026 Hard
More PYQs from JEE Mains
- Let \(p\) and \(q\) be two real numbers such that \(p+q=\) 3 and \(p^{4}+q^{4}=369\). Then \(\left(\frac{1}{p}+\frac{1}{q}\right)^{-2}\) is equal toJEE Mains 2022 Hard
- Let \(\overrightarrow{\mathrm{a}}=2 \hat{i}-\hat{j}+3 \hat{k}, \overrightarrow{\mathrm{~b}}=3 \hat{i}-5 \hat{j}+\hat{k}\) and \(\overrightarrow{\mathrm{c}}\) be a vector such that \(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}}\) and \((\vec{a}+\vec{c}) \cdot(\vec{b}+\vec{c})=168\). Then the maximum value of \(|\vec{c}|^2\) is :JEE Mains 2025 Medium
- Let \(f: R -\{3\} \rightarrow R -\{1\}\) be defined by \(f(x)=\frac{x-2}{x-3} .\) Let \(g: R \rightarrow R\) be given as \(g ( x )=2 x -3\). Then, the sum of all the values of \(x\) for which \(f^{-1}( x )+ g ^{-1}( x )=\frac{13}{2}\) is equal to ...... .JEE Mains 2021 Hard
- A hyperbola passes through the point \(P\left( {\sqrt 2 ,\sqrt 3 } \right)\) has foci at \(\left( { \pm 2,0} \right)\). Then the tangent to this hyperbola at \(P\) also passes through the pointJEE Mains 2017 Hard
- Let the point \((p, p+1)\) lie inside the region \(E=\left\{(x, y): 3-x \leq y \leq \sqrt{9-x^2}, 0 \leq x \leq 3\right\}\) If the set of all values of \(p\) is the interval \((a, b)\). then \(b^2+b-a^2\) is equal to \(.................\).JEE Mains 2023 Hard
- Let \(f\) and \(g\) be continuous functions on \([0, a]\) such that \(f(x) = f(a -x)\) and \(g(x) + g(a -x) = 4\), then \(\int\limits_0^a {f\left( x \right)g\left( x \right)dx} \) is equal toJEE Mains 2019 Hard