Calculating values gives 1, 2, 3, 5, 8, 13, 21, 34. Subtracting consecutive terms reveals Wn – Wn-1 = Wn-2, which means Wn = Wn-1 + Wn-2. This is the Virahanka-Fibonacci sequence.
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Writing x = ar³, y = ar⁹ and z = ar¹⁵, we compute y² = (ar⁹)² = a²r¹⁸. Also x x z = (ar³) x (ar¹⁵) = a²r¹⁸. Since y² = x x z, x, y, z form a GP.
The sum of k consecutive natural numbers is k(2a + k – 1) / 2 = 100, meaning k(2a + k – 1) = 200. Testing odd factors gives only k = 5, yielding 18 + 19 + 20 + ...
Since t7 – t5 = 12, we have 2d = 12, giving d = 6. With third term a + 2d = 16, substituting d gives a = 4. The required AP is 4, 10, 16, 22, ….
Starting with t1 equal to 2, the recursive rule gives the sequence 2, 4, 10, 28, 82, 244, 730. Therefore, 730 occurs at the seventh position. Hence, 730 is the seventh term of the sequence.