The coefficient of $x^{101}$ in the expression $(5+x)^{500}+x(5+x)^{499}+x^{2}(5+x)^{498}+\ldots . . x^{500}$, $\mathrm{x}>0$, is

(A) ${ }^{501} \mathrm{C}_{101}(5)^{399}$

(B) ${ }^{501} \mathrm{C}_{101}(5)^{400}$

(C) ${ }^{501} \mathrm{C}_{100}(5)^{400}$

(D) ${ }^{500} \mathrm{C}_{101}(5)^{399}$

JEE Main · 2022 · 25 Jun 2022 · Shift 2 · Binomial Theorem

The coefficient of

x^{101}

in the expression

(5 + x)^{500} + x (5 + x)^{499} + x^{2} (5 + x)^{498} + \dots .. x^{500}

x > 0

, is

Answer: (A)

The given expression is a geometric series with first term

(5 + x)^{500}

and common ratio

\frac{x}{5 + x}

S = \frac{( 5 + x ) ^{500} [ 1 - ( \frac{x}{5 + x} ) ^{501} ]}{1 - \frac{x}{5 + x}} = \frac{( 5 + x ) ^{501} - x ^{501}}{5}

The coefficient of

x^{101}

(5 + x)^{501}

(101 501) 5^{400}

.

The coefficient of

x^{101}

x^{501}

is 0.

Therefore, the coefficient of

x^{101}

in the given expression is:

\frac{( 101 501 ) 5 ^{400}}{5} = (101 501) 5^{399}

JEE Main · 2022 · 25 Jun 2022 · Shift 2 · Single correct option