, 2020

If the first positive integer in the list is $a$ and the second is $b$, then the third integer is $a+b$, the fourth is $b+(a+b)$ or $a+2b$, and the fifth is $(a+b)+(a+2b)$ or $2a+3b$.

Thus, we are asked to find the number of pairs of positive integers $a$ and $b$, where $a$ is less than $b$ (since the list is increasing), and for which $2a+3b=124$.

What is the largest possible value for $b$?

If $b=42$, then $3b=3\times42=126$ which is too large since $2a+3b=124$. (Note that a larger value of $b$ makes $3b$ even larger.)

If $b=41$, then $3b=3\times41=123$.

However in this case, we get that $2a=124-123=1$, which is not possible since $a$ is a positive integer.

If $b=40$, then $3b=3\times40=120$ and so $2a=4$ or $a=2$.

Thus, the largest possible value for $b$ is 40.

What is the smallest value for $b$?

If $b=26$, then $3b=3\times26=78$ and so $2a=124-78=46$ or $a=23$.

If $b=25$, then $3b=3\times25=75$.

However in this case, we get that $2a=124-75=49$, which is not possible since $a$ is a positive integer.

If $b=24$, then $3b=3\times24=72$ and so $2a=124-72=52$ or $a=26$.

However, if the first integer in the list is 26, then the second integer can not equal 24 since the list is increasing.

Smaller values of $b$ will give larger values of $a$, and so the smallest possible value of $b$ is 26.

From the values of $b$ attempted thus far, we notice that when $b$ is an odd integer, $3b$ is also odd (since the product of two odd integers is odd), and $124-3b$ is odd (since the difference between an even integer and an odd integer is odd).

So when $b$ is odd, $124-3b$ is odd, and so $2a$ is odd (since $2a=124-3b$).

However, $2a$ is even for every choice of the integer $a$ and so $b$ cannot be odd.

Conversely, when $b$ is even, $124-3b$ is even (as required), and so all even integer values of $b$ from 26 to 40 inclusive will satisfy the requirements.
These values of $b$ are $26,28,30,32,34,36,38,40$, and so there are 8 such lists of five integers that have 124 as the fifth integer.

Here are the 8 lists: $2,40, 42, 82, 124$; $5,38,43,81,124$; $8,36,44,80,124$; $11,34,45,79,124$; $14,32,46,78,124$; $17,30,47,77,124$; $20,28,48,76,124$; $23,26,49,75,124$.