63. A number series is formed according to a specific pattern: a², (5a + 2),…

A sequence in which every term is obtained by multiplying the previous term by the same fixed number is called a Geometric Progression (GP).

If the first term is A and this fixed multiplier (the common ratio) is r, the nth term is Tn = A × r(n − 1), so once A and r are known from a few terms, any far-away term — such as the 7th — can be found directly, without listing every term in between.

1. a is the smallest prime number, so a = 2.

2. Term 1 = a2 = 22 = 4.

3. Term 2 = 5a + 2 = 5 × 2 + 2 = 12.

4. Term 3 = (a + 4)2 = (2 + 4)2 = 62 = 36.

5. Term 4 = a2 × 33 = 4 × 27 = 108.

6. Check the ratio between consecutive terms: 12 ÷ 4 = 3, 36 ÷ 12 = 3, and 108 ÷ 36 = 3 — the ratio is constant, so this is a GP with first term A = 4 and common ratio r = 3.

7. General term: Tn = 4 × 3(n − 1). For the 7th term: T7 = 4 × 36 = 4 × 729 = 2916.

8. Required value = T7 ÷ 6 = 2916 ÷ 6 = 486.

Independent check: T7 is 3 steps beyond T4, so T7 = T4 × r3 = 108 × 27 = 2916, giving the same 2916 ÷ 6 = 486.

Hence, (7th term) ÷ 6 = 486.