Term of arithmetic progression with prime common differences

Let $d_1$ and $d_2$ be the common differences of series $a_n$ and $b_n$ respectively, where $d_1, d_2$ are prime numbers.

Given $b_2 = 0 \implies b_n = (n-2)d_2$. Thus, $b_9 = 7d_2$ and $b_{19} = 17d_2$.

We are given $a_5 = b_9 = 7d_2$ and $a_{19} = b_{19} = 17d_2$.

In series $a_n$: $a_{19} - a_5 = 14d_1 \implies 17d_2 - 7d_2 = 14d_1 \implies 10d_2 = 14d_1 \implies 5d_2 = 7d_1$

Since $d_1, d_2$ are prime numbers, $d_1 = 5$ and $d_2 = 7$.

Now, $a_{11}$ is the midpoint of $a_5$ and $a_{19}$: $a_{11} = \frac{a_5 + a_{19}}{2} = \frac{7d_2 + 17d_2}{2} = 12d_2 = 12 \times 7 = 84$