Select the number from among the given options that can replace the question mark (?) in the following series. 543, 518, 495, 474, 455, ?

Correct — B. Take first differences before looking for anything cleverer.

Rule: the amount subtracted falls by 2 each time — 25, 23, 21, 19, 17.

543 − 25 = 518
518 − 23 = 495
495 − 21 = 474

474 − 19 = 455
455 − 17 = 438 → option (b)

Why the others are wrong

• (a)
  444 — A gap of 11, not 17. That is the fourth difference down the chain, so option (a) runs three steps ahead of where the series has actually reached.
• (c)
  440 — A gap of 15, which is the difference due at the next step. Option (c) is one term too far along.
• (d)
  442 — A gap of 13, two steps ahead of the 17 that this term needs.

When the terms of a series fall by an uneven amount, write the gaps underneath before trying anything else. Here the gaps themselves form a tidy arithmetic run.

A series whose first differences change by a constant is quadratic in the term number, which is why squares, cubes and ratios all fail on it and simple subtraction succeeds.

The whole question is decided by one line of arithmetic once the differences are written down.

Key facts

• The first differences are 25, 23, 21, 19 and then 17, an arithmetic run falling by 2.
• When first differences are not constant, take second differences before testing squares or ratios.
• 455 − 17 = 438, which is the sixth term.

Common traps

• Subtracting 19 a second time instead of moving the difference on to 17
• Letting the difference fall by 2 twice in one step and subtracting 15
• Testing ratios or squares before taking the plain first differences

No directly related past PYQ was found.