Add the last two

The idea

A recursive rule may use two or more previous terms: the Virahānka–Fibonacci sequence has V₁ = 1, V₂ = 2 and Vₙ = Vₙ₋₁ + Vₙ₋₂ for n ≥ 3, each term being the sum of the previous two.

What comes after 8?

A recursive rule does not have to use only the one term before. It can use the previous two, or more. Look at this list: 1, 2, 3, 5, 8, …

What do you think comes next?

• 11
• 12
• 13

Add the previous two

The first two terms are given. Find each next term by adding the two terms before it.

Two terms back

A recursive rule may use two or more previous terms. The Virahānka–Fibonacci sequence has V₁ = 1, V₂ = 2 and Vₙ = Vₙ₋₁ + Vₙ₋₂ for n ≥ 3: each term is the sum of the previous two.

V₃ = V₂ + V₁ = 2 + 1 = 3, V₄ = V₃ + V₂ = 5, V₅ = V₄ + V₃ = 8. So the sequence is 1, 2, 3, 5, 8, 13, 21, 34, …

Virahānka first wrote it down explicitly in the 7th century CE, in his work *Vṛttajātisamuchaya*, in the context of Prakrit meter and poetry. Gopāla and Hemachandra studied it later, and then Fibonacci (c. 1200 CE).

Notes

A recursive rule may use two or more previous terms. The Virahānka–Fibonacci sequence: V₁ = 1, V₂ = 2 and Vₙ = Vₙ₋₁ + Vₙ₋₂ for n ≥ 3, each term the sum of the previous two.

Check yourself

In the Virahānka–Fibonacci sequence V₆ = 13 and V₇ = 21. What is V₈?

Answer: 34

V₈ = V₇ + V₆ = 21 + 13 = 34.

Which rule makes the Virahānka–Fibonacci terms?

Find V₉ in the Virahānka–Fibonacci sequence (V₇ = 21, V₈ = 34).

Answer: 55

V₉ = V₈ + V₇ = 34 + 21 = 55.

Which list starts 1, 2 and then makes each term the sum of the previous two?

• Vₙ = Vₙ₋₁ + Vₙ₋₂ — correct. Yes. Each term is the sum of the previous two, for n ≥ 3.
• Vₙ = 2 × Vₙ₋₁. Doubling gives 1, 2, 4, 8, … but the third term is 3 (2 + 1), not 4.
• Vₙ = Vₙ₋₁ + 1. Adding 1 gives 1, 2, 3, 4, … but the fourth term is 5 (3 + 2), not 4.
• 1, 2, 4, 8, 16, …. Here each term is double the one before. In our list the third term is 1 + 2 = 3.
• 1, 2, 3, 5, 8, 13, … — correct. Yes. 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13.
• 1, 2, 3, 4, 5, 6, …. The 4th term would be 2 + 3 = 5, not 4. Adding the previous two is not adding 1.