Select the missing number in the following sequence $3,6,24,30,63,72,?,?,195,210$
- $117,123$
- $120,132$
- $123,135$
- $135,144$
The sequence is pairs: (3,6), (24,30), (63,72), (120,132), (195,210). The pattern for the first number in each pair is n^2-1: 2^2-1=3, 5^2-1=24, 8^2-1=63, 11^2-1=120, 14^2-1=195. The second number is n^2+n: 2^2+2=6, 5^2+5=30, 8^2+8=72, 11^2+11=132, 14^2+14=210.
The series is built by adding successive squares to the previous terms: 3 plus 9 equals 12 is incorrect, but 3 plus 3 squared equals 12, and 12 plus 4 squared equals 28 is incorrect. The actual pattern involves adding the next odd number and then adding the reversed odd number: 3 plus 3 equals 6, 6 plus 18 equals 24, 24 plus 6 equals 30, 30 plus 33 equals 63, and 63 plus 9 equals 72, meaning this logic is wrong. The true pattern is adding consecutive perfect squares to the digits or terms: 3 plus 3 is 6, 6 plus 18 is 24, 24 plus 6 is 30, 30 plus 33 is 63, and 63 plus 9 is 72 is entirely wrong. The correct method is 3 plus 9 equals 12, but the series shows 6, so the real differences are 3, 18, 6, 33, 9, 48, 12, 63, and 15, where 3 plus 15 equals 18, 18 plus 15 equals 33, and 33 plus 15 equals 48, giving the sequence 3, 18, 33, 48, 63 and 6, 9, 12, 15, which adds to give the terms 72 plus 48 equals 120 and 120 plus 12 equals 132. The result is 120,132.