In a linear sequence, the 4th term and the 7th term are in the ratio 11:20. If the sum of the 4th and 7th terms is 124, find the sum of the first fifteen terms.

Let the first term be $a$ and the common difference be $d$.
The $n$-th term of a linear sequence is given by $a_n = a + (n-1)d$.
The 4th term is $a_4 = a + 3d$.
The 7th term is $a_7 = a + 6d$.

The 4th term and the 7th term are in the ratio $11:20$.

The sum of the 4th and 7th terms is 124.

Substitute $d = \frac{3}{2}a$ into the equation $2a + 9d = 124$:

Now find $d$:

So, the first term is $a=8$ and the common difference is $d=12$.

The sum of the first $n$ terms of an arithmetic sequence is $S_n = \frac{n}{2}(2a + (n-1)d)$.
For $n=15$:

The sum of the first fifteen terms is $\boxed{\text{1380}}$.

b)

Given the function $f:x \to 5x + 13$, we write it as $y = 5x + 13$.

To find the inverse function, we interchange $x$ and $y$:

Subtract 13 from both sides:

Divide by 5:

The inverse rule of the function is $\boxed{f^{-1}(x) = \frac{x - 13}{5}}$.