The coefficient of x ^ 101 in the expression (5 + x) ^ 500 + x (5 + x) ^ 499 + x ^ 2 (5 + x) ^ 498 + + x ^ 500 , x > 0 is

Correct answer: A. ^ 501 C _ 101 × 5 ^ 399 Given, (5 + x) ^ 500 + x (5 + x) ^ 499 + x ^ 2 (5 + x) ^ 498 + + x ^ 500 Using sum of the G.P in the given expression we get, ⇒ (5 + x) ^ 500 [ ((x)/(5 + x)) ^ 501 - 1] ((x)/(5 + x)) - 1 = x ^ 501 - (5 + x) ^ 501 - 5 = (1)/(5) (5 + x) ^ 501 - x ^ 501 5 Now coefficient of x ^ 101 in (1)/(5) (5 + x) ^ 501 will be (1)/(5)(C101 501) (5) ^ 501 - 101 = 5 ^ 399 (C101 501)

JEE Main 202225 JuneEvening Shift

Exam: JEE Main
Year: 2022
Date: 25 June
Shift: Evening Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2022 · 25 June · Evening Shift · Mathematics · Binomial Theorem

The coefficient of x ^ 101 in the expression (5 + x) ^ 500 + x (5 +...

The coefficient of

{x}^{101}

in the expression

{\left(5 + x\right)}^{500} + x{\left(5 + x\right)}^{499} + {x}^{2}{\left(5 + x\right)}^{498} + \ldots \ldots + {x}^{500}, x > 0

Options

A ${}^{501}{C}_{101} \times {5}^{399}$
B $501{C}_{101} \times {5}^{400}$
C $501{C}_{100} \times {5}^{400}$
D $500{C}_{101} \times {5}^{399}$

Correct answer

A. ${}^{501}{C}_{101} \times {5}^{399}$

Step-by-step solution

Given,

${\left(5 + x\right)}^{500} + x{\left(5 + x\right)}^{499} + {x}^{2}{\left(5 + x\right)}^{498} + \ldots \ldots + {x}^{500}$

Using sum of the G.P in the given expression we get,

$⇒ \dfrac{{\left(5 + x\right)}^{500}\left[{\left(\dfrac{x}{5 + x}\right)}^{501} - 1\right]}{\left(\dfrac{x}{5 + x}\right) - 1} = \dfrac{{x}^{501} - {\left(5 + x\right)}^{501}}{- 5}$

$= \dfrac{1}{5}{\left(5 + x\right)}^{501} - \dfrac{{x}^{501}}{5}$

Now coefficient of ${x}^{101}$ in $\dfrac{1}{5}{\left(5 + x\right)}^{501}$ will be $\dfrac{1}{5}\left(C101 501\right){\left(5\right)}^{501 - 101}$ $= {5}^{399}\left(C101 501\right)$

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