Select the number from among the given options that can replace the question mark (?) in the following series. 30, 22, 36, 75, 178, ?

Correct — A. Rule: multiply by 0.5, 1, 1.5, 2, 2.5 in turn, and add 7, 14, 21, 28, 35 — the multiplier grows by 0.5 and the addition by 7 at every step.

30 × 0.5 + 7 = 15 + 7 = 22
22 × 1 + 14 = 22 + 14 = 36
36 × 1.5 + 21 = 54 + 21 = 75
75 × 2 + 28 = 150 + 28 = 178

The next step continues both ladders:

178 × 2.5 + 35 = 445 + 35 = 480 → option (a)

Why the others are wrong

• (b)
  400 — 400 is short before the addition is even reached, since 178 × 2.5 is already 445. Holding the multiplier at 2 instead of advancing it gives 356 + 35 = 391, which is not 400 either.
• (c)
  361 — 361 is 19², and squaring is not what this line does. Every term here comes from a multiplier rising by 0.5 and an addition rising by 7.
• (d)
  343 — 343 is 7³, and it is smaller than 178 × 2 = 356. The multiplier only ever grows, so the next term cannot fall below twice the term before it.

Number series at this level are built either on a pattern in the differences or on a pattern in the operations, and the differences tell you which.

Here they run −8, +14, +39, +103. There is no arithmetic or geometric shape in that, and no clean second difference either, so the pattern is in the operations rather than the gaps.

Try a growing multiplier paired with a growing addition. When one candidate rule reproduces all five printed terms, that is far stronger evidence than a difference table ever gives.

Key facts

• The multipliers run 0.5, 1, 1.5, 2 and then 2.5, rising by 0.5 each step.
• The additions run 7, 14, 21, 28 and then 35, rising by 7 each step.
• 30 × 0.5 + 7 = 22, and 75 × 2 + 28 = 178, so the rule holds at both ends of the printed line.
• 178 × 2.5 + 35 = 480.

Common traps

• Reaching for a difference table first and abandoning the item when it looks random
• Assuming a fractional multiplier is impossible in a whole-number series
• Picking 361 or 343 because they are a neat square and cube, which no printed term is built on

No directly related past PYQ was found.