1B. The binomial coefficients of the second, third, and fourth terms in the expansion of the binomial $(1+x)^{n}$ form an arithmetic progression. Calculate $n$.
Solution. We have $\binom{n}{2}-\binom{n}{1}=\binom{n}{3}-\binom{n}{2}$. By rearranging, we get $n^{2}-9 n+14=0$. The solutions are $n_{1}=7$ and $n_{2}=2$, but for $n_{2}=2$ the problem does not make sense. Therefore, $n=7$.