This is an arithmetic sequence, and the task is to find the term index from a known term value. The catalog at the start of month n is an arithmetic sequence with first term a₁ = 420 and common difference d = 30, because 30 titles are added each month starting with month 2. The value of the nth term is aₙ = a₁ + (n − 1) × d, where the load-bearing piece is the (n − 1) factor: month 1 has had zero additions, month 2 has had one, and so on.
Set the term equal to the target: 420 + (n − 1) × 30 = 1,200. Subtract: (n − 1) × 30 = 780. Divide: n − 1 = 26. Add 1: n = 27. Check by back-substitution: month 27 gives 420 + 26 × 30 = 420 + 780 = 1,200, and month 26 gives 420 + 25 × 30 = 1,170, which is short, so 27 is the first month that reaches 1,200.
A quick back-substitution does not rescue a solver who used the wrong formula: the value 1,200 also appears at 420 + 26 × 30, so the flawed a₁ + n × d formula reproduces the target and simply attaches the index 26 to that same arithmetic, which is why pinning down the right index rule, not just checking the arithmetic, is the real point of the problem.