Q. What is X in the sequence 24, X, 12, 18, 36, 90?

Explanation :

1. Write down the six terms as
a₁ = 24,
a₂ = X,
a₃ = 12,
a₄ = 18,
a₅ = 36,
a₆ = 90.

2. Observe the multipliers from one term to the next (unknown for the first two steps):
a₁ → a₂ multiply by m₁
a₂ → a₃ multiply by m₂
a₃ → a₄ multiply by m₃ = 18/12 = 1.5
a₄ → a₅ multiply by m₄ = 36/18 = 2
a₅ → a₆ multiply by m₅ = 90/36 = 2.5

3. Notice m₃, m₄, m₅ form an arithmetic sequence increasing by 0.5:
m₃ = 1.5 = 3·0.5,
m₄ = 2.0 = 4·0.5,
m₅ = 2.5 = 5·0.5.
It is natural to extend backward to
m₂ = 2·0.5 = 1.0,
m₁ = 1·0.5 = 0.5.

4. Compute a₂ and check a₃:
a₂ = a₁·m₁ = 24·0.5 = 12,
a₃ should be a₂·m₂ = 12·1.0 = 12,
which matches the given third term.

Hence X = a₂ = 12.