The AP of multiples of (12) greater than (400) is (408,420,432,\ldots). What is its (18)th term?

Correct answer: (612)

Direct answer: Option A, 612. An arithmetic progression, or AP, is a list in which the same number is added each time. The first multiple of 12 greater than 400 is 408, because 12 × 33 = 396 and 12 × 34 = 408. Thus the first term is a = 408 and the common difference is d = 12. The nth-term rule is \(a_n=a+(n-1)d\). For the 18th term, \(a_{18}=408+(18-1)\times12=408+17\times12=408+204=612\). Checking the pattern also gives 408 as the first term and 612 after seventeen equal jumps of 12. Option A is correct because it is exactly the 18th term. Option B, 624, is the 19th term, since it is one further jump of 12. Option C, 636, is the 20th term. Option D, 648, is the 21st term. All are multiples of 12, but they come too late in the sequence. Memory cue: for the nth AP term, count n minus 1 jumps, not n jumps.