Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202125 JulyMorning Shift
Exam
JEE Main
Year
2021
Date
25 July
Shift
Morning Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2021 · 25 July · Morning Shift · Mathematics · Binomial Theorem

If b is very small as compared to the value of a , so that the cube...

If bb is very small as compared to the value of aa, so that the cube and other higher powers of ba\dfrac{b}{a} can be neglected in the identity

1a−b+1a−2b+1a−3b+….+1a−nb=\alphan+βn2+γn3\dfrac{1}{a - b} + \dfrac{1}{a - 2b} + \dfrac{1}{a - 3b} + \ldots . + \dfrac{1}{a - nb} = \alphan + \beta{n}^{2} + \gamma{n}^{3}

then the value of γ\gamma is :

Options

  1. Aa2+b3a3\dfrac{{a}^{2} + b}{3{a}^{3}}
  2. Ba+b3a2\dfrac{a + b}{3{a}^{2}}
  3. Cb23a3\dfrac{{b}^{2}}{3{a}^{3}}
  4. Da+b23a3\dfrac{a + {b}^{2}}{3{a}^{3}}

Correct answer

C. b23a3\dfrac{{b}^{2}}{3{a}^{3}}

Step-by-step solution

Given,

1a−b+1a−2b+1a−3b+….+1a−nb\dfrac{1}{a - b} + \dfrac{1}{a - 2b} + \dfrac{1}{a - 3b} + \ldots . + \dfrac{1}{a - nb}

=(a−b)−1+(a−2b)−1+….+(a−nb)−1= {\left(a - b\right)}^{- 1} + {\left(a - 2b\right)}^{- 1} + \ldots . + {\left(a - nb\right)}^{- 1}

=1a∑r=1n(1−rba)−1= \dfrac{1}{a}\mathop{∑}_{r = 1}^{n}{\left(1 - \dfrac{rb}{a}\right)}^{- 1}

=1a∑r=1n{(1+rba+r2b2a2)+(termstobeneglected)}= \dfrac{1}{a}\mathop{∑}_{r = 1}^{n}\left\{\left(1 + \dfrac{rb}{a} + \dfrac{{r}^{2}{b}^{2}}{{a}^{2}}\right) + \left(termstobeneglected\right)\right\}

=1a[n+n(n+1)2⋅ba+n(n+1)(2n+1)6⋅b2a2]= \dfrac{1}{a}\left[n + \dfrac{n ( n + 1 )}{2} \cdot \dfrac{b}{a} + \dfrac{n ( n + 1 ) ( 2n + 1 )}{6} \cdot \dfrac{{b}^{2}}{{a}^{2}}\right]

=1a[n+n2+n2⋅ba+2n3+3n2+n6⋅b2a2]= \dfrac{1}{a}\left[n + \dfrac{{n}^{2} + n}{2} \cdot \dfrac{b}{a} + \dfrac{2{n}^{3} + 3{n}^{2} + n}{6} \cdot \dfrac{{b}^{2}}{{a}^{2}}\right]

So γ=b23a3\gamma = \dfrac{{b}^{2}}{3{a}^{3}}

Practice Binomial Theorem on Quantrex Academy →

More from Binomial Theorem

If 26((2^3)/(3) 12 2 + (2^5)/(5) 12 4 + (2^7)/(7) 12 6 + … + 2^ 13 13 12 12 ) = 3^ 13 - α… JEE Main 2026If (1 - x^3)^ 10 = ∑ _ r=0 ^ 10 a_r x^r (1-x)^ 30-2r , then 9a_9 a_ 10 is equal to… JEE Main 2026If the coefficients of the middle terms in the binomial expansions of (1 + α x)^ 26 and… JEE Main 2026The coefficient of x^2 in the expansion of (2x^2 + (1)/(x))^ 10 , x ≠ 0 , is : JEE Main 2026If the sum of the coefficients of x^7 and x^ 14 in the expansion of ((1)/(x^3) - x^4)^n… JEE Main 2026In the expansion of (9x- 1 3 x )^ 18 , x>0 , if the term independent of x is (221)k… JEE Main 2026Let the smallest value of k ∈ ℕ , for which the coefficient of x^3 in (1+x)^3 + (1+x)^4 +… JEE Main 2026If for 3 ≤ r ≤ 30 , 30 30-r + 3 30 31-r + 3 30 32-r + 30 33-r = m r , then m equals: JEE Main 2026The number of elements in the set S = \ (r, k) : k ∈ ℤ and ^ 36 C_ r+1 = 6( ^ 35 C_r)… JEE Main 2026Given below are two statements : Statement I: 25^ 13 +20^ 13 +8^ 13 +3^ 13 is divisible… JEE Main 2026The sum of the coefficients of x^ 499 and x^ 500 in (1+x)^ 1000 +x(1+x)^ 999 +x^ 2 (1+x)^… JEE Main 2026Let S=(1)/(25!)+(1)/(3!23!)+(1)/(5!21!)+… up to 13 terms. If 13 ~S= 2^ k n! , k ∈ ~N… JEE Main 2026The value of ^ 100 C_ 50 51 + ^ 100 C_ 51 52 +….+ ^ 100 C_ 100 101 is : JEE Main 2026The sum of all possible values of n ∈ N , so that the coefficients of x, x^ 2 and x^ 3 in… JEE Main 2026Let C_ r denote the coefficient of x^ r in the binomial expansion of (1+x)^ n , n ∈ N, 0… JEE Main 2026The coefficient of x^ 48 in (1+x)+2(1+x)^ 2 +3(1+x)^ 3 +…+100(1+x)^ 100 is equal to JEE Main 2026 Full Binomial Theorem list Binomial Theorem JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs