Binomial Theorem · Sum of binomial coefficients

Let the smallest value of

k \in \mathbb{N}

, for which the coefficient of

x^3

(1 + x)^3 + (1 + x)^4 + (1 + x)^5 + \ldots + (1 + x)^{99} + (1 + kx)^{100}

x \neq 0

, is

\left(43n + \dfrac{101}{4}\right)\left({}^{100}C_3\right)

for some

n \in \mathbb{N}

, be

p

. Then the value of

p + n

is:

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