Determine Parameter in Binomial Expansion Coefficient Relation

To find the value of $n$, we first determine the coefficient of $x^2$ in the given algebraic expansion: $S = (1+x)^2 + (1+x)^3 + \dots + (1+x)^{49} + (1+mx)^{50}$

The coefficient of $x^2$ in $(1+x)^k$ is given by $\binom{k}{2}$. Thus, the coefficient of $x^2$ in the first part of the expression is: $\sum_{k=2}^{49} \binom{k}{2}$

Using the Hockey-stick identity, $\sum_{k=r}^{n} \binom{k}{r} = \binom{n+1}{r+1}$, we get: $\sum_{k=2}^{49} \binom{k}{2} = \binom{50}{3}$

The coefficient of $x^2$ in $(1+mx)^{50}$ is: $\binom{50}{2} m^2$

Therefore, the total coefficient of $x^2$ in $S$ is: $C = \binom{50}{3} + m^2 \binom{50}{2}$

We can express $\binom{50}{3}$ and $\binom{50}{2}$ in terms of $\binom{51}{3}$: $\binom{51}{3} = \frac{51 \times 50 \times 49}{3 \times 2 \times 1} = 17 \times \frac{50 \times 49}{2} = 17 \binom{50}{2}$ $\implies \binom{50}{2} = \frac{1}{17} \binom{51}{3}$

Using Pascal's Identity $\binom{51}{3} = \binom{50}{3} + \binom{50}{2}$, we have: $\binom{50}{3} = \binom{51}{3} - \binom{50}{2} = \left(1 - \frac{1}{17}\right) \binom{51}{3} = \frac{16}{17} \binom{51}{3}$

Substituting these back into the expression for $C$: $C = \frac{16}{17} \binom{51}{3} + \frac{m^2}{17} \binom{51}{3} = \left(\frac{m^2 + 16}{17}\right) \binom{51}{3}$

We are given that this coefficient is equal to $(3n + 1) \binom{51}{3}$. Equating the two expressions: $\frac{m^2 + 16}{17} = 3n + 1$ $m^2 + 16 = 51n + 17$ $m^2 - 1 = 51n$

For $n$ to be a positive integer, $m^2 - 1$ must be a positive multiple of $51 = 3 \times 17$. Thus: $(m-1)(m+1) \equiv 0 \pmod{51}$

This implies that:

1. $m \equiv \pm 1 \pmod 3$
2. $m \equiv \pm 1 \pmod{17}$

Checking the smallest positive integers $m > 1$ satisfying $m \equiv \pm 1 \pmod{17}$:

• For $m = 16$: $m^2 - 1 = 16^2 - 1 = 255$ Since $255 = 51 \times 5$, $m = 16$ makes $n = 5$, which is a positive integer.

Since $m = 16$ is the smallest positive integer ($m > 1$) satisfying the given condition, the corresponding value of $n$ is: $n = \frac{16^2 - 1}{51} = \frac{255}{51} = 5$