The value of (1)/(1 ! 50 !) + (1)/(3 ! 48 !) + (1)/(5 ! 46 !) + . + (1)/(49 ! 2 !) + (1)/(51 ! 1 !) is

Correct answer: B. 2 ^ 50 51 ! Let S = (1)/(1 ! · 50 !) + (1)/(3 ! · 48 !) + (1)/(5 ! · 46 !) + . . . . + (1)/(51 ! · 1 !) ⇒ S = (1)/(51 !)[(51 !)/(1 ! · 50 !) + (51 !)/(3 ! · 48 !) + (51 !)/(5 ! · 46 !) + . . . . + (51 !)/(51 ! · 0 !)] ⇒ S = (1)/(51 !)[C1 51 + C3 51 + C5 51 + . . . . + C51 51] ⇒ S = 2 ^ 51 - 1 51 ! ⇒ S = 2 ^ 50 51 !

JEE Main 20231 FebruaryMorning Shift

Exam: JEE Main
Year: 2023
Date: 1 February
Shift: Morning Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2023 · 1 February · Morning Shift · Mathematics · Binomial Theorem

The value of (1)/(1 ! 50 !) + (1)/(3 ! 48 !) + (1)/(5 ! 46 !) + . +...

The value of

$\dfrac{1}{1 ! 50 !} + \dfrac{1}{3 ! 48 !} + \dfrac{1}{5 ! 46 !} + \ldots . + \dfrac{1}{49 ! 2 !} + \dfrac{1}{51 ! 1 !}$ is

Options

A $\dfrac{{2}^{50}}{50 !}$
B $\dfrac{{2}^{50}}{51 !}$
C $\dfrac{{2}^{51}}{51 !}$
D $\dfrac{{2}^{51}}{50 !}$

Correct answer

B. $\dfrac{{2}^{50}}{51 !}$

Step-by-step solution

Let

$S = \dfrac{1}{1 ! \cdot 50 !} + \dfrac{1}{3 ! \cdot 48 !} + \dfrac{1}{5 ! \cdot 46 !} + . . . . + \dfrac{1}{51 ! \cdot 1 !}$

$⇒ S = \dfrac{1}{51 !}\left[\dfrac{51 !}{1 ! \cdot 50 !} + \dfrac{51 !}{3 ! \cdot 48 !} + \dfrac{51 !}{5 ! \cdot 46 !} + . . . . + \dfrac{51 !}{51 ! \cdot 0 !}\right]$

$⇒ S = \dfrac{1}{51 !}\left[C1 51 + C3 51 + C5 51 + . . . . + C51 51\right]$

$⇒ S = \dfrac{{2}^{51 - 1}}{51 !}$

$⇒ S = \dfrac{{2}^{50}}{51 !}$

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