If the bottom row of an row pyramid contains the first Virahāṅka-Fibonacci numbers, what can we say about the numbers in the pyramid? What can we say about the number at the top?

Question

Homi · Accepted Answer

We will build the pyramid using the given rules and look for patterns in the numbers.

Step 1 — Understanding Virahāṅka-Fibonacci Numbers

Virahāṅka-Fibonacci numbers are a special sequence of numbers. Each number is the sum of the two numbers before it.

Let $F_k$ be the $k$ -th Virahāṅka-Fibonacci number.

The first two numbers are both 1. So, we have: $F_1 = 1$ $F_2 = 1$

Then, we find the next numbers by adding the previous two: $F_3 = F_2 + F_1$ $F_3 = 1 + 1$ $= 2$

$F_4 = F_3 + F_2$ $F_4 = 2 + 1$ $= 3$

$F_5 = F_4 + F_3$ $F_5 = 3 + 2$ $= 5$

$F_6 = F_5 + F_4$ $F_6 = 5 + 3$ $= 8$

$F_7 = F_6 + F_5$ $F_7 = 8 + 5$ $= 13$

$\boxed{\text{The sequence starts: } 1, 1, 2, 3, 5, 8, 13, \dots}$

Step 2 — Building Pyramids for Small $n$

A pyramid is built such that each number is the sum of the two numbers directly below it. The bottom row has $n$ Virahāṅka-Fibonacci numbers.

Let us look at small values of $n$ .

For $n=1$ : The bottom row has $F_1$ . The pyramid has only one number.

$F_1$

$\boxed{\text{Top number for } n=1 \text{ is } F_1}$

For $n=2$ : The bottom row has $F_1, F_2$ . We know $F_1=1$ and $F_2=1$ . The row above has one number, which is $F_1 + F_2$ .

$F_1 + F_2$ $= 1 + 1$ $= 2$

We know $F_3 = 2$ .

$F_3$ $F_1 \quad F_2$

$\boxed{\text{Top number for } n=2 \text{ is } F_3}$

For $n=3$ : The bottom row has $F_1, F_2, F_3$ . We know $F_1=1, F_2=1, F_3=2$ . The row above the bottom row will have two numbers.

The first number is $F_1 + F_2$ . $F_1 + F_2 = 1 + 1 = 2$ This is $F_3$ .

The second number is $F_2 + F_3$ . $F_2 + F_3 = 1 + 2 = 3$ This is $F_4$ .

So, the second row from the bottom is $F_3, F_4$ . The top row will have one number, which is the sum of these two.

$F_3 + F_4$ $= 2 + 3$ $= 5$

We know $F_5 = 5$ .

$F_5$ $F_3 \quad F_4$ $F_1 \quad F_2 \quad F_3$

$\boxed{\text{Top number for } n=3 \text{ is } F_5}$

For $n=4$ : The bottom row has $F_1, F_2, F_3, F_4$ . Numbers are $1, 1, 2, 3$ .

The second row from the bottom: $F_1+F_2 = 1+1=2 = F_3$ $F_2+F_3 = 1+2=3 = F_4$ $F_3+F_4 = 2+3=5 = F_5$ So, this row is $F_3, F_4, F_5$ .

The third row from the bottom: $F_3+F_4 = 2+3=5 = F_5$ $F_4+F_5 = 3+5=8 = F_6$ So, this row is $F_5, F_6$ .

The top row (fourth row from bottom): $F_5+F_6 = 5+8=13 = F_7$ So, the top number is $F_7$ .

$F_7$ $F_5 \quad F_6$ $F_3 \quad F_4 \quad F_5$ $F_1 \quad F_2 \quad F_3 \quad F_4$

$\boxed{\text{Top number for } n=4 \text{ is } F_7}$

Diagram 1

Step 3 — Generalizing the Pattern for All Numbers in the Pyramid

Let us observe the numbers in each row. The bottom row (Row 1 from bottom) is $F_1, F_2, F_3, \dots, F_n$ . The next row (Row 2 from bottom) is formed by adding adjacent numbers. $F_1+F_2 = F_3$ $F_2+F_3 = F_4$ $F_3+F_4 = F_5$ So, Row 2 is $F_3, F_4, F_5, \dots, F_{n+1}$ .

The next row (Row 3 from bottom) is formed similarly. $F_3+F_4 = F_5$ $F_4+F_5 = F_6$ So, Row 3 is $F_5, F_6, F_7, \dots, F_{n+2}$ .

We can see a clear pattern. Each number in the pyramid is a Virahāṅka-Fibonacci number. If a row contains consecutive Virahāṅka-Fibonacci numbers, say $F_k, F_{k+1}, F_{k+2}, \dots$ , then the row above it will contain $F_k+F_{k+1}, F_{k+1}+F_{k+2}, \dots$ . Using the definition $F_m = F_{m-1} + F_{m-2}$ , we know $F_k+F_{k+1} = F_{k+2}$ . So, the row above will contain $F_{k+2}, F_{k+3}, F_{k+4}, \dots$ . This means all numbers in every row of the pyramid will always be Virahāṅka-Fibonacci numbers.

Step 4 — Finding the Number at the Top

Let us look at the first number in each row, starting from the bottom. Row 1 (bottom row) starts with $F_1$ . Row 2 starts with $F_3$ . Row 3 starts with $F_5$ . Row 4 starts with $F_7$ .

We can see that for the $k$ -th row from the bottom, the first number is $F_{2k-1}$ . The pyramid has $n$ rows. So, the top row is the $n$ -th row from the bottom. For the $n$ -th row, the first number will be $F_{2n-1}$ .

Also, we notice that the number of elements in each row decreases by one as we go up. Row 1 has $n$ numbers. Row 2 has $n-1$ numbers. Row 3 has $n-2$ numbers. The $n$ -th row (the top row) will have $n-(n-1) = 1$ number. Since the top row has only one number, that number must be the first number of that row.

So, the number at the top of the pyramid is $F_{2n-1}$ .

Answer

(i) All the numbers in the pyramid are Virahāṅka-Fibonacci numbers. (ii) The number at the top of the pyramid is the (2n - 1)th Virahāṅka-Fibonacci number.

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