JEE Mains · Maths · STD 11 - 8. sequence and series
Let for \(n =1,2, \ldots \ldots, 50, S _{ a }\) be the sum of the infinite geometric progression whose first term is \(n ^{2}\) and whose common ratio is \(\frac{1}{(n+1)^{2}}\). Then the value of \(\frac{1}{26}+\sum\limits_{n=1}^{50}\left(S_{n}+\frac{2}{n+1}-n-1\right)\) is equal to
- A \(41600\)
- B \(47651\)
- C \(41651\)
- D \(41671\)
Answer & Solution
Correct Answer
(C) \(41651\)
Step-by-step Solution
Detailed explanation
\(S_{n}=\frac{n^{2}}{1-\frac{1}{(n+1)^{2}}}=\frac{n(n+1)^{2}}{(n+2)}\) \(S_{n}=\frac{n\left(n^{2}+2 n+1\right)}{(n+2)}\) \(S_{n}=\frac{n[n(n+2)+1]}{(n+2)}\) \(S_{n}=n\left[n+\frac{1}{n+2}\right]\) \(S_{n}=n^{2}+\frac{n+2-2}{(n+2)}\) \(S_{n}=n^{2}+1-\frac{2}{(n+2)}\) Now…
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 straight lines \(l_1\) and \(l_2\) pass through the origin and trisect the line segment of the line \(L: 9 x+5 y=\) 45 between the axes. If \(m_1\) and \(m_2\) are the slopes of the lines \(l_1\) and \(1_2\),then the point of intersection of the line \(y =\left( m _1+ m _2\right) x\) with \(L\) lies onJEE Mains 2023 Hard
- A vector \(\overrightarrow{ V }\) in the first octant is inclined to the \(x\) axis at \(60^{\circ}\), to the \(y\)-axis at \(45^{\circ}\) and to the z-axis at an acute angle. If a plane passing through the points \((\sqrt{2},-1,1)\) and \(( a , b , c )\), is normal to \(\overrightarrow{ v }\), thenJEE Mains 2023 Hard
- The shortest distance between the lines \(\frac{x+2}{1}=\frac{y}{-2}=\frac{z-5}{2}\) and \(\frac{x-4}{1}=\frac{y-1}{2}=\frac{z+3}{0}\) is \(......\).JEE Mains 2023 Hard
- If the domain of the function \(f ( x )=\frac{[ x ]}{1+ x ^2}\), where \([x]\) is greatest integer \(\leq x\), is \((2,6)\), then its range isJEE Mains 2023 Hard
- The greatest integer less than or equal to the sum of first \(100\) terms of the sequence \(\frac{1}{3}, \frac{5}{9}, \frac{19}{27}, \frac{65}{81}, \ldots \ldots\) is equal toJEE Mains 2022 Hard
- If \(\mathrm{S}=\{\mathrm{a} \in \mathrm{R}:|2 \mathrm{a}-1|=3[\mathrm{a}]+2\{\mathrm{a}\}\}\), where \([\mathrm{t}]\) denotes the greatest integer less than or equal to \(t\) and \(\{t\}\) represents the fractional part of \(t\), then \(72 \sum_{\mathrm{a} \in \mathrm{S}} \mathrm{a}\) is equal to ....................JEE Mains 2024 Hard
More PYQs from JEE Mains
- Let \(A=\left[a_{i j}\right], a_{i j} \in Z \cap[0,4], 1 \leq i, j \leq 2\). The number of matrices \(A\) such that the sum of all entries is a prime number \(p \in(2,13)\) is \(........\).JEE Mains 2023 Hard
- If \(\int \limits_{-0.15}^{0.15}\left|100 x ^2-1\right| dx =\frac{ k }{3000}\), then \(k\) is equal to \(..........\).JEE Mains 2023 Hard
- Let \(S_{n}\) be the sum of the first \(n\) terms of an arithmetic progression. If \(S_{3 n}=3 S_{2 n}\), then the value of \(\frac{S_{4 n}}{S_{2 n}}\) is:JEE Mains 2021 Medium
- The sum of the following series \(1 + 6 + \frac{{9({1^2} + {2^2} + {3^2})}}{7} + \frac{{12({1^2} + {2^2} + {3^2} + {4^2})}}{9} + \frac{{15({1^2} + {2^2} + .... + {5^2})}}{{11}} + ...\) up to \(15\) terms, isJEE Mains 2019 Hard
- The radius of circle, having minimum area, which touches the cruve \(y = 4 - {x^2}\) and the lines \(y = \left| x \right|\) is :JEE Mains 2017 Hard
- If \(f\left( x \right) = \left[ x \right] - \left[ {\frac{x}{4}} \right],\,x \in R\) , where \([x]\) denotes the greatest integer function, thenJEE Mains 2019 Hard