Question

Find the missing term in the series: \[ 0,\;1,\;3,\;2,\;8,\;6,\;15,\;24,\;\_\_\_,\;120 \]

• A. \(21\)
• B. \(23\)
• C. \(24\)
• D. \(25\)

Hint:

If a series looks irregular, split it into odd and even positions. Many times two separate patterns exist.

Answer 1

The Correct Option is C

Concept: Observe the series in alternating positions. Split into two separate sequences.

Step 1: Odd position terms.
Odd-position terms: \[ 0,\;3,\;8,\;15,\;\_\_ \] Pattern: \[ 0=1^2-1 \] \[ 3=2^2-1 \] \[ 8=3^2-1 \] \[ 15=4^2-1 \] Next: \[ 5^2-1=24 \]

Step 2: Even position terms.
Even-position terms: \[ 1,\;2,\;6,\;24,\;120 \] Pattern: \[ 1!=1 \] \[ 2!=2 \] \[ 3!=6 \] \[ 4!=24 \] \[ 5!=120 \] This confirms the alternating pattern. Thus the missing term is: \[ \boxed{24} \]