Equivalence of Binomial Series

Prove that $7^n\left\{1+\frac{n}{7}+\frac{n(n-1)}{7\times 14}+\frac{n(n-1)(n-2)}{7\times 14\times 21}+\dots \right\}=4^n\left\{1+\frac{n}{2}+\frac{n(n+1)}{2\times 4}+\frac{n(n+1)(n+2)}{2\times 4\times 6}+\dots \right\}$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XIV. Binomial Theorem: Any Index, Examples XIV.c, No. 8