Question 71
$$
\sum_{j=0}^{n} \left\{ \binom{3n+2-j}{j} 2^j - \binom{3n+1-j}{j-1} 2^{j-1} \right\} = 2^{3n}
$$
confirm that.
$$
\text{Here } \binom{n}{k} = C_n^k \text{ and } j = 0 \text{ when } \binom{3n+1-j}{j-1} 2^{j-1} = 0 \text{ considered.}
$$
\sum_{j=0}^{n} \left\{ \binom{3n+2-j}{j} 2^j - \binom{3n+1-j}{j-1} 2^{j-1} \right\} = 2^{3n}
$$
confirm that.
$$
\text{Here } \binom{n}{k} = C_n^k \text{ and } j = 0 \text{ when } \binom{3n+1-j}{j-1} 2^{j-1} = 0 \text{ considered.}
$$