JEE Mains · Maths · STD 11 - 8. sequence and series
If the sum of the first 20 terms of the series
\(\frac{4.1}{4+3.1^2+1^4}+\frac{4.2}{4+3.2^2+2^4}+\frac{4.3}{4+3.3^2+3^4}+\frac{4.4}{4+3.4^2+4^4}+\ldots\)
is \(\frac{m}{n}\), where \(m\) and \(n\) are coprime, then \(m+n\) is equal to :-
- A \(423\)
- B \(420\)
- C \(421\)
- D \(422\)
Answer & Solution
Correct Answer
(C) \(421\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \sum_{\mathrm{r}=1}^{20} \frac{4 \mathrm{r}}{4+3 \mathrm{r}^2+\mathrm{r}^4} \\ & \sum_{\mathrm{r}=1}^{20} \frac{4 \mathrm{r}}{\left(\mathrm{r}^2+\mathrm{r}+2\right)\left(\mathrm{r}^2-\mathrm{r}+2\right)} \\ & 2…
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 total number of \(5\)-digit numbers, formed by using the digits \(1,2,3,5,6,7\) without repetition, which are multiple of \(6\), isJEE Mains 2022 Medium
- Let the normal at the point \(P\) on the parabola \(y ^{2}=\) \(6 x\) pass through the point \((5,-8)\). If the tangent at \(P\) to the parabola intersects its directrix at the point \(Q\), then the ordinate of the point \(Q\) isJEE Mains 2022 Medium
- If \(y = {e^{nx}}\), then \(\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)\left( {\frac{{{d^2}x}}{{d{y^2}}}} \right)\) is equal toJEE Mains 2014 Hard
- Let \(\mathrm{n}\) be a non-negative integer. Then the number of divisors of the form " \(4 \mathrm{n}+1\) " of the number \((10)^{10} \cdot(11)^{11} \cdot(13)^{13}\) is equal to \(....\)JEE Mains 2021 Hard
- Two parabolas with a common vertex and with axes along \(x-\) axis and \(y-\) axis, respectively, intersect each other in the first quadrant. if the length of the latus rectum of each parabola is \(3\) , then the equation of the common tangent to the two parabolas is?JEE Mains 2018 Hard
- If \(z\) is a complex number such that \(\frac{z-i}{z-1}\) is purely imaginary, then the minimum value of \(\mid \mathrm{z}-(3+3 \mathrm{i}) \mid\) is :JEE Mains 2021 Hard
More PYQs from JEE Mains
- Consider an arithmetic series and a geometric series having four initial terms from the set \(\{11,8,21,16,26,32,4\}\) If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to .......JEE Mains 2021 Medium
- If \(x^2 + y^2 + sin\, y = 4\), then the value of \(\frac{{{d^2}y}}{{d{x^2}}}\) at the point \((- 2, 0)\) isJEE Mains 2018 Hard
- Two circles each of radius \(5\, units\) touch each other at the point \((1,2)\). If the equation of their common tangent is \(4 \mathrm{x}+3 \mathrm{y}=10\), and \(\mathrm{C}_{1}(\alpha, \beta)\) and \(\mathrm{C}_{2}(\gamma, \delta)\), \(\mathrm{C}_{1} \neq \mathrm{C}_{2}\) are their centres, then \(|(\alpha+\beta)(\gamma+\delta)|\) is equal to .... .JEE Mains 2021 Hard
- Let \(\vec{a}\) and \(\vec{b}\) be the vectors of the same magnitude such that \(\frac{|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}|-|\vec{a}-\vec{b}|}=\sqrt{2}+1\). Then \(\frac{|\vec{a}+\vec{b}|^2}{|\vec{a}|^2}\) is :JEE Mains 2025 Medium
- Let \( f(x) = \begin{cases} \frac{ax^{2}+2ax+3}{4x^{2}+4x-3}, & x \neq -\frac{3}{2}, \frac{1}{2} \\ b, & x = -\frac{3}{2}, \frac{1}{2} \end{cases} \) be continuous at \( x=-\frac{3}{2} \). If \( fof(x) = \frac{7}{5} \), then \( x \) is equal to:JEE Mains 2026 Medium
- The sum of squares of all the real solutions of the equation \(\log_{(x+1)}(2x^2+5x+3) = 4 - \log_{(2x+3)}(x^2+2x+1)\) is equal to ________.JEE Mains 2026 Medium