Question

Find the missing terms in the series: \[ 225,\;4,\;196,\;16,\;169,\;36,\;\_\_\_,\;\_\_\_,\;121,\;100,\;100,\;144,\;81,\;196 \]

• A. \(160,30\)
• B. \(154,44\)
• C. \(150,54\)
• D. \(144,64\)

Hint:

If a long series looks irregular, try splitting into odd and even positions. Many times each follows its own clear pattern.

Answer 1

The Correct Option is D

Concept: Separate the series into two alternating sequences. One sequence contains decreasing perfect squares. The other contains increasing perfect squares.

Step 1: Odd-position terms.
Odd-position terms: \[ 225,\;196,\;169,\;\_\_,\;121,\;100,\;81 \] These are: \[ 15^2,\;14^2,\;13^2,\;12^2,\;11^2,\;10^2,\;9^2 \] So the missing odd term: \[ 12^2=144 \]

Step 2: Even-position terms.
Even-position terms: \[ 4,\;16,\;36,\;\_\_,\;100,\;144,\;196 \] These are: \[ 2^2,\;4^2,\;6^2,\;8^2,\;10^2,\;12^2,\;14^2 \] So the missing even term: \[ 8^2=64 \]

Step 3: Write the missing terms.
Thus: \[ 144,\;64 \] Hence, the required answer is: \[ \boxed{144,\;64} \]