If (4, k, 14, 19, ...) is an arithmetic progression, what is the value of k?

Correct answer: 9

Answer: 9, so option D is correct. In an arithmetic progression, the difference between every two consecutive terms is the same. The known consecutive terms 14 and 19 allow us to find the common difference: d = 19 - 14 = 5. The term immediately before 14 must therefore be 5 less than 14, because adding 5 to it must produce 14. Hence k = 14 - 5 = 9. We can verify the complete beginning of the progression: 4, 9, 14, 19, ... . Its consecutive differences are 9 - 4 = 5, 14 - 9 = 5, and 19 - 14 = 5, so the defining condition is satisfied. Option A, 6, would give differences 2 and 8 around k; option B, 7, would give 3 and 7; and option C, 8, would give 4 and 6. None of these has equal consecutive differences. A useful memory cue is: in an AP, move backward by subtracting the common difference and move forward by adding it.