Lilavati (Bhaskaracharya)

The Sum of Squares and Cubes

Rule:
Twice the number of terms increased by one, divided by three, and then multiplied by the Summation Saṅkalita: the sum of the first 'n' natural numbers, or $\frac{n(n+1)}{2}$, is the Sum of Squares original: kṛti-yogaḥ; the formula is $\frac{2n+1}{3} \times \frac{n(n+1)}{2} = \frac{n(n+1)(2n+1)}{6}$.

The Sum of Cubes original: ghanaikyam starting from one is equal to the square of that same Summation. 3

Statement of the Problem:
O mathematician, if your mind is skilled in the path of summation, tell me quickly the Sum of Squares original: vargaikam and the Sum of Cubes original: dhanaikam of those same numbers Referring to the sequence from 1 to 7 used in the previous section. 4

Numerical Setup:

The result:
The Sum of Squares of these is 140.
The Sum of Cubes is 784 calculated as $28^2$, where 28 is the sum of numbers 1 through 7.

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General Arithmetic Progressions

Rule:
The number of terms minus one, multiplied by the common difference pracaya: the constant amount added to each term and added to the first term mukha: literally 'face' or 'mouth', meaning the starting value, results in the last term original: antya-dhanaṃ.

That last term, added to the first term and divided by two, is the middle value original: madhya-dhanaṃ. That middle value, multiplied by the number of terms, is the total sum original: sarva-dhanaṃ of the series. 5

Statement of the Problem:
A man gave two The verse says "two" (dvau), but the subsequent data entry uses "four" (ādi 4) drammas dramma: a silver coin of the medieval period to learned ones on the first day. He continued to give daily, friend, with an increase of five each day. Tell me, how many drammas were given in total when fifteen days had passed? 6

Numerical Setup:

Example:
In a case where the first term is seven, the common difference is five, and the number of terms is eight; tell me the middle value, the last term, and the total sum.