Aryabhatt’s – Sum of Sums

– Chandrahas M. Halai

This is in continuation of my earlier article:

https://chandrahasblogs.wordpress.com/2021/07/30/how-many-laddoos/

On the first day, I gifted my beloved

A Diamond.

On the second day, I gifted my beloved

Two Emeralds

And A Diamond.

On the third day, I gifted my beloved

Three Rubies

Two Emeralds

And A Diamond.

On the fourth day, I gifted my beloved

Four Sapphires

Three Rubies

Two Emeralds

And A Diamond.

On the fifth day, I gifted my beloved

Five Pearls

Four Sapphires

Three Rubies

Two Emeralds

And A Diamond.

On the sixth day, I gifted my beloved

Six Jades

Five Pearls

Four Sapphires

Three Rubies

Two Emeralds

And A Diamond.

On the seventh day, I gifted my beloved

Seven Opals

Six Jades

Five Pearls

Four Sapphires

Three Rubies

Two Emeralds

And A Diamond.

On the eighth day, I gifted my beloved

Eight Gold coins

Seven Opals

Six Jades

Five Pearls

Four Sapphires

Three Rubies

Two Emeralds

And A Diamond.

On the ninth day, I gifted my beloved

Nine Silver coins

Eight Gold coins

Seven Opals

Six Jades

Five Pearls

Four Sapphires

Three Rubies

Two Emeralds

And A Diamond.

Tell me how many jewels I gifted to my beloved?

Note: I am still single. 🙂

The sum we want is

In general, we can write this as

In sigma (summation) notation, we can write this as:

Thus the total number of gifts that I gave to my beloved is:

This formula was derived by Aryabhatt. He used to call the sum of sums as वारसंकलित​ (Vaarasankalita). The formulas for the sums of higher orders were given by Narayan Pandit in his work Ganitakaumudi (composed in 1356 CE).

These numbers are called Tetrahedral numbers as these many number of things can be arranged into a Tetrahedron (Triangular pyramid). Refer the picture and diagram given below.

Image courtesy: Wikimedia commons

If we observe the picture we see that the stack is made up of layers of triangular number of marbles. That is, tetrahedral numbers are sums of triangular numbers. The n^th tetrahedral number is the sum of first n triangular numbers.

The formulas for the sums of higher orders derived by Narayan Pandit will be discussed in the next article.

How Many Laddoos?

– Chandrahas M. Halai

Can you quickly tell me how many laddoos are there in a stack?

If we observe the picture we see that the stack is made up of layers of square number of laddoos. Let us say that the bottom most layer contains n² laddoos. Then the next layer will contain (n-1)² laddoos, the next will have (n-2)² and so on. Hence the total number of laddoos will be

This is the sum of squares of n natural numbers. These are square pyramidal numbers. How do you calculate these sums?

Let us begin by calculating the sum of first n natural numbers.

Now, let us write this sum in reverse order as given below:

Let us now add both the equations, we get

Hence, the sum of natural numbers from 1 to 100 will be

These are called the triangular numbers as these many numbers of things can be arranged in to a triangle. Refer diagram 1. Here, 1, 3, 6, 10 and 15 are triangular numbers.

Diagram 1

Now, how do we calculate the square pyramidal number? For this, let us consider:

Let us add up all the above equations:

This formula gives us the square pyramidal number.

If there are 8 layers in the stack of laddoos, then we have

Hence, there will be 204 laddoos in a 8 layered stack.

Now, let us consider a sum of cubes of first n natural numbers.

Let

We can use a method similar to that which we used to get the sum of squares of n natural numbers. Thus, we have

Let us add up all the above equations:

The sum of cubes of first 7 natural numbers is:

Aryabhatt had derived the above three formulas. He had called these sums of powers of natural numbers as संकलित (Sankalita). संकलित means addition.

Aryabhatt was the first to derive the formula for the sum of cubes of natural numbers.