Question 149
Find the greatest positive real number $M$ such that for all positive real sequence $\left(a_{n}\right)$ and for all real number $m \lt M$, it is possible to find some index $n \geq 1$ that satisfies the inequality
$$
a_{1} + a_{2} + a_{3} + \cdots + a_{n} + a_{n+1} \gt m a_{n}.
$$
$$
a_{1} + a_{2} + a_{3} + \cdots + a_{n} + a_{n+1} \gt m a_{n}.
$$