Arithmetic sequence, finding 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.
Why the wrong choices tempt: (A) 26 is the dominant trap, landing on (1,200 − 420)/30 = 26 because the solver uses a₁ + n × d and drops the (n − 1) correction, forgetting that month 1 carries no additions. (C) 28 over-corrects the same index in the opposite direction by counting the starting catalog as month 0 and still adding 1. (D) 40 comes from ignoring the starting 420 and treating 1,200/30 as a simple rate from zero, mistaking the running catalog total for the monthly addition rate. (E) 52 halves the common difference, dividing the 780-title gap by 15 instead of 30, a difference-versus-ratio mix-up. Note that 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 same number 26 surfaces under the wrong index, which is why the off-by-one in the index, not the arithmetic, is the real point of the problem.