Select the number from among the given options that can replace the question mark (?) in the following series. 949, 1006, 1066, 1129, 1195, ?

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

949 → 1006 is +57
1006 → 1066 is +60
1066 → 1129 is +63
1129 → 1195 is +66

Rule: the difference itself rises by 3 each step, so the next gap is 66 + 3 = 69.

1195 + 69 = 1264 → option (a).

Why the others are wrong

• (b)
  1233 — 1233 is 1195 + 38, a gap smaller than the 66 before it. Every gap so far has grown, so any answer below 1195 + 66 = 1261 is out on sight.
• (c)
  1254 — 1254 is 1195 + 59. That would make the difference fall from 66 to 59 on the last step alone, after four steps of steady growth.
• (d)
  1244 — 1244 is 1195 + 49. Like the other near-misses it needs the run of differences to reverse direction, which nothing in the series supports.

For a numeric series, write the first differences under the terms before you do anything else. They are 57, 60, 63 and 66.

Those rise by a constant 3, so the second difference is 3. A series with a constant second difference is quadratic in its position, and you never need to find the quadratic — you only need to continue the differences.

Extend the difference row by one term, 66 + 3 = 69, and add it to the last printed term. That is 1195 + 69 = 1264.

Key facts

• The first differences of 949, 1006, 1066, 1129 and 1195 are 57, 60, 63 and 66.
• Those differences rise by a constant 3, so the second difference is 3.
• The next difference is 69, giving 1195 + 69 = 1264.

Common traps

• Reading it as percentage growth. The ratios drift from 1.060 down to 1.058, so no constant ratio fits even though one nearly does.
• Reusing the last difference instead of growing it, which gives 1195 + 66 = 1261 and is not on the list.
• Mis-subtracting one gap. 1129 − 1066 = 63, and a slip there moves the predicted answer by 3.

No directly related past PYQ was found.