UFM Additional Further Pure

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jee-advanced 2013 Q47 Evaluation of a Finite or Infinite Sum View
The value of $\cot \left( \sum _ { n = 1 } ^ { 23 } \cot ^ { - 1 } \left( 1 + \sum _ { k = 1 } ^ { n } 2 k \right) \right)$ is
(A) $\frac { 23 } { 25 }$
(B) $\frac { 25 } { 23 }$
(C) $\frac { 23 } { 24 }$
(D) $\frac { 24 } { 23 }$
jee-advanced 2013 Q51 Evaluation of a Finite or Infinite Sum View
Let $S _ { n } = \sum _ { k = 1 } ^ { 4 n } ( - 1 ) ^ { \frac { k ( k + 1 ) } { 2 } } k ^ { 2 }$. Then $S _ { n }$ can take value(s)
(A) 1056
(B) 1088
(C) 1120
(D) 1332
jee-main 2012 Q64 Evaluation of a Finite or Infinite Sum View
The sum of the series $$\frac{1}{1+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{4}} + \ldots$$ upto 15 terms is
(1) 1
(2) 2
(3) 3
(4) 4
jee-main 2012 Q65 Evaluation of a Finite or Infinite Sum View
The sum of the series $1 ^ { 2 } + 2.2 ^ { 2 } + 3 ^ { 2 } + 2.4 ^ { 2 } + 5 ^ { 2 } + 2.6 ^ { 2 } + \ldots . + 2 ( 2 m ) ^ { 2 }$ is
(1) $m ( 2 m + 1 ) ^ { 2 }$
(2) $m ^ { 2 } ( m + 2 )$
(3) $m ^ { 2 } ( 2 m + 1 )$
(4) $m ( m + 2 ) ^ { 2 }$
jee-main 2012 Q65 Evaluation of a Finite or Infinite Sum View
The sum of the series $1 + \frac { 4 } { 3 } + \frac { 10 } { 9 } + \frac { 28 } { 27 } + \ldots$ upto $n$ terms is
(1) $\frac { 7 } { 6 } n + \frac { 1 } { 6 } - \frac { 2 } { 3.2 ^ { n - 1 } }$
(2) $\frac { 5 } { 3 } n - \frac { 7 } { 6 } + \frac { 1 } { 2.3 ^ { n - 1 } }$
(3) $n + \frac { 1 } { 2 } - \frac { 1 } { 2 \cdot 3 ^ { n } }$
(4) $n - \frac { 1 } { 3 } - \frac { 1 } { 3.2 ^ { n - 1 } }$
jee-main 2012 Q65 Evaluation of a Finite or Infinite Sum View
If the sum of the series $1 ^ { 2 } + 2 \cdot 2 ^ { 2 } + 3 ^ { 2 } + 2 \cdot 4 ^ { 2 } + 5 ^ { 2 } + \ldots 2.6 ^ { 2 } + \ldots$ upto n terms, when n is even, is $\frac { n ( n + 1 ) ^ { 2 } } { 2 }$, then the sum of the series, when n is odd, is
(1) $n ^ { 2 } ( n + 1 )$
(2) $\frac { n ^ { 2 } ( n - 1 ) } { 2 }$
(3) $\frac { n ^ { 2 } ( n + 1 ) } { 2 }$
(4) $n ^ { 2 } ( n - 1 )$
jee-main 2013 Q63 Evaluation of a Finite or Infinite Sum View
The sum of the series : $( 2 ) ^ { 2 } + 2 ( 4 ) ^ { 2 } + 3 ( 6 ) ^ { 2 } + \ldots$ upto 10 terms is :
(1) 11300
(2) 11200
(3) 12100
(4) 12300
jee-main 2013 Q65 Evaluation of a Finite or Infinite Sum View
The sum of the series: $1 + \frac { 1 } { 1 + 2 } + \frac { 1 } { 1 + 2 + 3 } + \ldots\ldots$. upto 10 terms, is:
(1) $\frac { 18 } { 11 }$
(2) $\frac { 22 } { 13 }$
(3) $\frac { 20 } { 11 }$
(4) $\frac { 16 } { 9 }$
jee-main 2013 Q65 Evaluation of a Finite or Infinite Sum View
The value of $1^{2} + 3^{2} + 5^{2} + \cdots + 25^{2}$ is:
(1) 2925
(2) 1469
(3) 1728
(4) 1456
jee-main 2016 Q62 Evaluation of a Finite or Infinite Sum View
If the sum of the first ten terms of the series $\left(1\frac{3}{5}\right)^{2}+\left(2\frac{2}{5}\right)^{2}+\left(3\frac{1}{5}\right)^{2}+4^{2}+\left(4\frac{4}{5}\right)^{2}+\ldots$, is $\frac{16}{5} m$, then $m$ is equal to: (1) 102 (2) 101 (3) 100 (4) 99
jee-main 2019 Q65 Evaluation of a Finite or Infinite Sum View
Let $S _ { k } = \frac { 1 + 2 + 3 + \ldots + k } { k }$. If $S _ { 1 } ^ { 2 } + S _ { 2 } ^ { 2 } + \ldots + S _ { 10 } ^ { 2 } = \frac { 5 } { 12 } A$, then $A$ is equal to :
(1) 301
(2) 303
(3) 156
(4) 283
jee-main 2020 Q64 Recurrence Relations and Sequence Properties View
Let $f : R \rightarrow R$ be a function which satisfies $f ( x + y ) = f ( x ) + f ( y ) , \forall x , y \in R$. If $f ( 1 ) = 2$ and $g ( n ) = \sum _ { k = 1 } ^ { ( n - 1 ) } f ( k ) , n \in N$ then the value of $n$, for which $g ( n ) = 20$, is
(1) 5
(2) 20
(3) 4
(4) 9
jee-main 2021 Q62 Evaluation of a Finite or Infinite Sum View
If $n \geqslant 2$ is a positive integer, then the sum of the series ${ } ^ { n + 1 } C _ { 2 } + 2 \left( { } ^ { 2 } C _ { 2 } + { } ^ { 3 } C _ { 2 } + { } ^ { 4 } C _ { 2 } + \ldots + { } ^ { n } C _ { 2 } \right)$ is
(1) $\frac { n ( n - 1 ) ( 2 n + 1 ) } { 6 }$
(2) $\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$
(3) $\frac { n ( n + 1 ) ^ { 2 } ( n + 2 ) } { 12 }$
(4) $\frac { n ( 2 n + 1 ) ( 3 n + 1 ) } { 6 }$
jee-main 2021 Q62 Evaluation of a Finite or Infinite Sum View
The sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } + 6 n + 10 } { ( 2 n + 1 ) ! }$ is equal to
(1) $\frac { 41 } { 8 } e + \frac { 19 } { 8 } e ^ { - 1 } + 10$
(2) $\frac { 41 } { 8 } e + \frac { 19 } { 8 } e ^ { - 1 } - 10$
(3) $- \frac { 41 } { 8 } e + \frac { 19 } { 8 } e ^ { - 1 } - 10$
(4) $\frac { 41 } { 8 } e - \frac { 19 } { 8 } e ^ { - 1 } - 10$
jee-main 2021 Q62 Evaluation of a Finite or Infinite Sum View
The sum of the series $\frac { 1 } { x + 1 } + \frac { 2 } { x ^ { 2 } + 1 } + \frac { 2 ^ { 2 } } { x ^ { 4 } + 1 } + \ldots + \frac { 2 ^ { 100 } } { x ^ { 2 ^ { 100 } } + 1 }$ when $x = 2$ is:
(1) $1 - \frac { 2 ^ { 101 } } { 4 ^ { 101 } - 1 }$
(2) $1 + \frac { 2 ^ { 101 } } { 4 ^ { 101 } - 1 }$
(3) $1 + \frac { 2 ^ { 100 } } { 4 ^ { 101 } - 1 }$
(4) $1 - \frac { 2 ^ { 100 } } { 4 ^ { 201 } - 1 }$
jee-main 2021 Q62 Evaluation of a Finite or Infinite Sum View
If $0 < x < 1$ and $y = \frac { 1 } { 2 } x ^ { 2 } + \frac { 2 } { 3 } x ^ { 3 } + \frac { 3 } { 4 } x ^ { 4 } + \ldots \ldots$, then the value of $e ^ { 1 + y }$ at $x = \frac { 1 } { 2 }$ is: (1) $\frac { 1 } { 2 } e ^ { 2 }$ (2) $2 e$ (3) $2 e ^ { 2 }$ (4) $\frac { 1 } { 2 } \sqrt { \mathrm { e } }$
jee-main 2021 Q63 Evaluation of a Finite or Infinite Sum View
Let $S _ { n } = 1 \cdot ( n - 1 ) + 2 \cdot ( n - 2 ) + 3 \cdot ( n - 3 ) + \ldots + ( n - 1 ) \cdot 1 , \quad n \geqslant 4$. The sum $\sum _ { n = 4 } ^ { \infty } \frac { 2 S _ { n } } { n ! } - \frac { 1 } { ( n - 2 ) ! }$ is equal to :
(1) $\frac { e - 2 } { 6 }$
(2) $\frac { e - 1 } { 3 }$
(3) $\frac { e } { 6 }$
(4) $\frac { \mathrm { e } } { 3 }$
jee-main 2021 Q65 Evaluation of a Finite or Infinite Sum View
$\frac { 1 } { 3 ^ { 2 } - 1 } + \frac { 1 } { 5 ^ { 2 } - 1 } + \frac { 1 } { 7 ^ { 2 } - 1 } + \ldots + \frac { 1 } { ( 201 ) ^ { 2 } - 1 }$ is equal to
(1) $\frac { 101 } { 404 }$
(2) $\frac { 25 } { 101 }$
(3) $\frac { 101 } { 408 }$
(4) $\frac { 99 } { 400 }$
jee-main 2021 Q73 Evaluation of a Finite or Infinite Sum View
If $\cot ^ { - 1 } ( \alpha ) = \cot ^ { - 1 } 2 + \cot ^ { - 1 } 8 + \cot ^ { - 1 } 18 + \cot ^ { - 1 } 32 + \ldots$ upto 100 terms, then $\alpha$ is:
(1) 1.01
(2) 1.00
(3) 1.02
(4) 1.03
jee-main 2021 Q73 Evaluation of a Finite or Infinite Sum View
If $[ x ]$ be the greatest integer less than or equal to $x$, then $\sum _ { n = 8 } ^ { 100 } \left[ \frac { ( - 1 ) ^ { n } n } { 2 } \right]$ is equal to:
(1) 0
(2) 4
(3) - 2
(4) 2
jee-main 2022 Q61 Evaluation of a Finite or Infinite Sum View
If $A = \sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( 3 + ( - 1 ) ^ { n } \right) ^ { n } }$ and $B = \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \left( 3 + ( - 1 ) ^ { n } \right) ^ { n } }$, then $\frac { A } { B }$ is equal to
(1) $\frac { 11 } { 9 }$
(2) 1
(3) $- \frac { 11 } { 9 }$
(4) $- \frac { 11 } { 3 }$
jee-main 2022 Q62 Evaluation of a Finite or Infinite Sum View
The sum $\sum_{n=1}^{21} \frac{3}{(4n-1)(4n+3)}$ is equal to
(1) $\frac{7}{87}$
(2) $\frac{7}{29}$
(3) $\frac{14}{87}$
(4) $\frac{21}{29}$
jee-main 2023 Q62 Evaluation of a Finite or Infinite Sum View
If $a_n = \frac{-2}{4n^2 - 16n + 15}$, then $a_1 + a_2 + \ldots + a_{25}$ is equal to:
(1) $\frac{51}{144}$
(2) $\frac{49}{138}$
(3) $\frac{50}{141}$
(4) $\frac{52}{147}$
jee-main 2023 Q63 Evaluation of a Finite or Infinite Sum View
If $S _ { n } = 4 + 11 + 21 + 34 + 50 + \ldots$ to $n$ terms, then $\frac { 1 } { 60 } \left( S _ { 29 } - S _ { 9 } \right)$ is equal to
(1) 223
(2) 226
(3) 220
(4) 227
jee-main 2023 Q64 Evaluation of a Finite or Infinite Sum View
If $\frac { 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } \ldots\ldots \text { upto } n \text { terms } } { 1 \cdot 3 + 2 \cdot 5 + 3 \cdot 7 + \ldots.. \text { upto } n \text { terms } } = \frac { 9 } { 5 }$ then the value of $n$ is

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