The missing number in the series 40, 120, 60, 180, 90, ?, 135 is

Concept: An alternating-operation series is built by applying two different rules turn by turn — one rule at every odd-numbered step and a second rule at every even-numbered step. To expose such a pattern, compare each term with the one immediately before it and record the factor by which it grows or shrinks; if those step-factors repeat with a period of two, the series is alternating, and the missing term is produced by whichever of the two rules falls due at that position.

Application: read off the step-factor of each consecutive pair in 40, 120, 60, 180, 90, ?, 135.

1. 40 → 120: 120 = 40 × 3, so the first step multiplies by 3.

2. 120 → 60: 60 = 120 ÷ 2, so the second step divides by 2.

3. 60 → 180: 180 = 60 × 3, so the third step multiplies by 3.

4. 180 → 90: 90 = 180 ÷ 2, so the fourth step divides by 2.

5. The two rules alternate with period two, so the fifth step must multiply by 3: 90 × 3 = 270.

The complete rule table for the series is therefore:

Cross-check: the sixth step must then divide by 2, and 270 ÷ 2 = 135 — exactly the last term printed in the series. The alternating cycle closes on the data the question already supplies, which independently confirms 270.

Contrast with the other values offered:

• 110 — halving it gives 55, not the 135 that follows, and a ×3 step on 90 gives 270, not 110.

• 105 — halving it gives 52.5, which cannot produce the whole number 135.

• 210 — halving it gives 105, not 135, and a ×3 step on 90 gives 270, not 210.