Q1.
[Virahānka–Fibonacci sequence 1, 2, 3, 5, 8, 13, 21, 34, …] Can you write the next two terms of this sequence?
Answer
Use the recursive rule Vn = Vn–1 + Vn–2, adding the last two terms each time.
V8 = 34, V7 = 21
V9 = V8 + V7 = 34 + 21 = 55
V10 = V9 + V8 = 55 + 34 = 89
V9 = V8 + V7 = 34 + 21 = 55
V10 = V9 + V8 = 55 + 34 = 89
The sequence continues 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
Why it happens: this rule reaches back two steps, not one, so it needs two starting values — here V1 = 1 and V2 = 2. That is why the rule is only stated for n ≥ 3: for n = 1 or n = 2 there simply are not two earlier terms to add. A recursive rule is complete only when its starting terms are given along with it.
Did you know? Virahānka wrote this sequence down in the 7th century CE in the Vṛttajātisamuchaya, while counting the ways of arranging short and long syllables in Prakrit verse. Gopāla and Hemachandra studied it in the 12th century — about fifty years before Fibonacci met it.