Number Play - Nature's Favourite Sequence: The Virahāṅka–Fibonacci Numbers

Find the $10^{th}$ term of the Virahāṅka–Fibonacci sequence if the sequence starts with $F_1 = 1$ and $F_2 = 1$.

The sequence is: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55$. The $10^{th}$ term is $55$.

We use the rule $F_n = F_{n-1} + F_{n-2}$. $F_3 = 1+1 = 2$, $F_4 = 2+1 = 3$, $F_5 = 3+2 = 5$, $F_6 = 5+3 = 8$, $F_7 = 8+5 = 13$, $F_8 = 13+8 = 21$, $F_9 = 21+13 = 34$, $F_{10} = 34+21 = 55$.

Calculate the sum of the first $6$ terms of the sequence: $1, 1, 2, 3, 5, 8$.

$1 + 1 + 2 + 3 + 5 + 8 = 20$

Adding the terms vertically: $\begin{array}{r} 1 \\ 1 \\ 2 \\ 3 \\ 5 \\ + 8 \\ \hline 20 \end{array}$ Alternatively, using the formula for the sum of $n$ terms: $F_{n+2} - 1$. Here $n=6$, so $F_{6+2} - 1 = F_8 - 1 = 21 - 1 = 20$.

If the $12^{th}$ term is $144$ and the $13^{th}$ term is $233$, find the $14^{th}$ term.

$F_{14} = 377$

Using the relation $F_n = F_{n-1} + F_{n-2}$: $F_{14} = F_{13} + F_{12}$ $F_{14} = 233 + 144 = 377$