- The 288
^{th}term of the series*a, b, b, c, c, c, d, d, d, d, e, e, e, e, e, …*is?

- There are 8436 steel balls, each with a radius of 1cm, stacked in a pile, with 1 ball on top, 3 balls in the second layer, 6 balls in the third, 10 in the fourth and so on. Find the number of horizontal layers in the pile?

**Interesting and challenging problems, aren’t they?**

The above two problems were asked in CAT, the entrance exams for admission into India’s most prestigious Business schools “The IIMs”. In these exams you are expected to solve each problem in less than two minutes. These questions are multiple choice questions. But, you get negative marks for the wrong choice.

So, how do you solve such problems not only accurately but also quickly?

## Let us begin by solving the first problem.

We have one ‘*a*’, two ‘*b*’s, three ‘*c*’s, four ‘*d*’s, and so on. These can be arranged in the shape of a triangle. Hence, it is obvious we are dealing with triangular numbers. For more on triangular numbers refer chapter 11 “*How Many Spheres?*” from my new book “*Meru Prastaar*”. Available at https://garudabooks.com/meru-prastaar

**Triangular Numbers**

We have to find the row number n for the 288^{th} term. Hence, we have

Triangular numbers are given by the formula:

Thus, we have

and

We have,

Thus the 288^{th} term lies in the 24^{th} row. And the 24^{th} character in the English alphabets is ‘*x*’. Hence, the 288^{th} term in the given series will be ‘*x*’.

## Now, let us solve the second problem

We have 1 ball on top, 3 balls in the second layer, 6 balls in the third, 10 in the fourth and so on. Now 1, 3, 6 and 10 are triangular numbers. Hence, here we have a stack of triangles. And a stack of triangles is a tetrahedron. It is now obvious that 8436 is a tetrahedral number. To understand tetrahedral numbers read chapter 12 “*Aryabhata’s Sum of Sums*” from my new book “*Meru Prastaar*”.

**Tetrahedral Numbers**

The *n*th tetrahedral number is given by:

Hence, we have

We have

Hence, we have *n* = 36

Thus, we will have 36 horizontal layers of triangles.

“**Meru Prastaar can Bell the CAT**”.

Chandrahas Halai is an Engineer, Scientist, Mathematician, Author, Photographer, Painter, and Designer. He is an engineering, Computer, and management consultant interested in diverse fields like mathematics, physics, aerospace, mechatronics, and mechanical engineering. His research papers and articles have been in reputed journals. He is also the author of the book ‘Vedic Mathematics Inside Out’.