I initially approached this problem by just playing some games with the computer and tried to recognise some patterns. I started by playing first but then swapped to the computer playing first. It was here that I noticed that it would always start with the same first number depending on the target number.
I noted this down and also noted that it would try to reach certain values consistently. I called these "landing values" and wrote them down.
From here, I noticed that there was a fixed distance between each landing value that was 1 greater than the largest number I could choose. This revelation led to further investigations that led to the process below.
For anyone who wanted to consider solving this problem, or ones like it, I think the best first step is to just play around and consider different values. After you've considered different values, you can assess whether there is a pattern or not. This pattern is the real starting off point for finding a general solution.
Finalised Process:
The numbers that are chosen to be added up range from 1 -> n, where n is a positive integer and the spaces between 1 and "n" are integers. For example, "1, 2, 3, 4" where n is 4.
The player will have a target number, which we will call "m". To begin, player 1 will write down the target and subtract (n+1) away from the number "m" until there is a number that they are able to choose, we will call this "b". Consider the target is 22(m) and the player can add up to 4(n), they will subtract 5(n+1) until they reach the number 2(b).
Player 1 will choose this number and begin the game. From this point forward, it is Player 1's objective to reach the next multiple of (n+1) that starts from "b". In our example, these "Landing Values" would be 7, 12, 17 and finally, 22. Player 1 will always be able to force a Landing Value on their turn assuming they started from "b" and their opponent plays perfectly.
If player 1 aims for these numbers, Player 2's choice will not matter as they will end up unable to reach the target number.
The exception to this trend comes if the target number is a multiple of (n+1). This is because you can't keep subtracting (n+1) to reach the initial number "b" as you will eventually reach (n+1) and the next number would be 0. For example, if we could choose to add up numbers between 1 and 4, if the target was 10, we could subtract 5(n+1) once. Following this, subtracting 5 again would reach 0, which isn't an option with our initial conditions. Thus, the first player would have to choose any number between 1 and 4.
Assuming that there is a perfect opponent, they would reach the multiple of (n+1) instead, in this example it would be 5. The game would continue as stated previously, resulting in player 2 winning instead.
Thus, you can reach the conclusion that it is only sensible to be Player 1 if the target number is not a multiple of (n+1), if it is, then you should choose to be Player 2.