Select the number from among the given options that can replace the question mark (?) in the following series. 5, 10, 17, 26, 37, ?

Correct — D. Rule: the differences run +5, +7, +9, +11, +13 — each gap is 2 more than the last.

10 - 5 = 5
17 - 10 = 7
26 - 17 = 9
37 - 26 = 11

The next gap is 13, so 37 + 13 = 50, which is option (d).

The same list read another way: each term is n squared plus 1, with n running 2, 3, 4, 5, 6. The sixth term is therefore 7 squared plus 1 = 50.

Why the others are wrong

• (a)
  54 — 54 would need a gap of 17, which is two rungs further up the ladder than the +13 the pattern calls for.
• (b)
  52 — 52 would need a gap of 15. That is the gap after the right one, so this is the answer for a candidate who counted one step too far.
• (c)
  60 — 60 would need a gap of 23, which no continuation of a ladder rising by 2 from 11 ever reaches.

A number series with unequal gaps is solved by taking the first differences and reading them as a series in their own right.

Here the first differences are 5, 7, 9 and 11, themselves an arithmetic run with a constant second difference of 2. A constant second difference always means the terms fit a quadratic, which is why the same list is also n squared plus 1.

Both readings earn their keep. The difference ladder is faster under exam pressure, and the quadratic form catches an arithmetic slip in the ladder.

Key facts

• First differences of the series: 5, 7, 9, 11, then 13.
• A constant second difference means the terms are a quadratic in the position number.
• The terms are n squared plus 1 for n = 2 to 7, giving 5, 10, 17, 26, 37, 50.

Common traps

• Taking one difference and assuming the gap is constant.
• Adding the last difference again, which gives 37 + 11 = 48.
• Continuing the ladder one rung too far and answering 52.

No directly related past PYQ was found.