And now I begin...
The Līlavātī gives the standard method of solving a quadratic equation by completing the squares, and gives the formula for the general solution. For a quadratic equation ax2+bx+c = 0, the general solution is
After describing this method and giving the above formula, Bhāskara has set out several exercises for the student to practice. One such exercise, based on the Sanskrit epic Mahābhārata, is
A topic taught in high school is the topic of arithmetic and geometric progressions, and related results such as the sum of the first n whole numbers, the sum of the squares of the first n whole numbers, and the sum of the cubes of the first n whole numbers. An arithmetic progression is a sequence of numbers such that the difference between any two consecutive numbers is a fixed amount, this fixed amount being called the common difference, while in a geometric progression it the ratio between any two consecutive numbers of the sequence that is fixed, and which is called the common ratio. All the basic results of arithmetic and geometric progressions taught in high school were well known to Indian mathematicians. For example, the Aryabhaṭia of Aryabhaṭa gives the sum of a given number of terms of an arithmetic progression, how to find the number of terms of an arithmetic progression given the sum, etc. Bhāskara revises all these results in the chapter Shredhivyavahara in the Līlavātī. In particular, he gives the following results (also given in Aryabhaṭia):
Regardinggeometric progressions. Bhāskara gives a method to find the sum of a given number of terms of a geometric progression in the chapter Shredhivyavaharaha from Līlavātī:
Typical problems in combinatorics at the high school level involve permutations and combinations, i.e. given n objects, in how many ways can we select r objects out of them? If only the number of combinations, and not the respective permutations in each of these combinations, is required, then the answer is denoted by nCr (C for combinations), and if the respective permutations are also required, then the answer is denoted by nPr.
In addition, he also gives the result for the number of permutations of n objects in the following verse in the Līlavātī, chapter ankapāsha:
“To find the number of permutations of given (n) different digits (or objects), write 1 in the first place, 2, 3, 4, . . . up to the number of objects (n) and multiply them. (This is the first part). Divide the product of the number of permutations and the sum of the given (n) digits by the number of the given digits (i.e. by n); write the quotient the given number of times (i.e. n times) in a column but leaving one-digit place each time; add them; the result is the sum of the numbers formed (by permuting the given n digits). (This is the second part.) Using (i) 2, 8, (ii) 3, 8, 9, (iii) 2, 3, . . . 9 how many different numbers can be formed? What is the sum of numbers, so formed, in each case?” (This is the third part.)
If the permutations nPr of r objects from n objects is required, it is easy to see, using the formula for nCr, along with the above result, that the answer is simply nCr times r! i.e.
"Find (i) 5+0 (ii) the square of 0, the cube of 0, the square root of 0, the cube root of 0, (iii) 0x5 (iv) 10/0, (v) A certain number is multiplied by zero and added to half the result. If the sum so obtained is first multiplied by three and then divided by zero, the result is 63. Find the original number."
The fifth case is of interest here. Looking at the earlier verses in the chapter, it is clear that Bhāskara is intuitively referring to what is called a limit. Using the rules from his earlier verses, the conditions in (v) can be expressed in the more familiar terminology of limits as:
(So, although the number of beings is changing during universal creation and dissolution, the one independent Reality, which is what Lord Sri Krishna is an Avatar of, is unchanging. This explains the seeming contradiction in verses 4 and 5 since, if Sri Krishna was in those beings, which are subject to birth and death, then He would also be subject to birth and death. On the other hand, He, as the supreme independent Reality, has to be in them, since those beings, subject to birth and death, cannot have an independent existence. So, although the number of beings is changing, Lord Krishna (or Lord Vishnu) is not changing, which is what Bhāskara is trying to explain by comparing khahara with Lord Vishnu.)
 André Weil, Number Theory, an approach through history from Hammurapi to Legendre, Birkhäuser 2007.