Lilavati (Bhaskaracharya)

20

Circle 1 ādi: first term 1 caya: common difference 5 gaccha: number of terms 8 sarvadhana: total sum 148 antyadhana: last term 36.

In this case, when the number of terms original: gaccha is an even number, there is no single middle day. Therefore, one must understand that the "middle value" is derived from the average of the two middle terms. To find the first term original: ādi, the following rule original: sūtra is given:

Rule for Finding the First Term

The total sum divided by the number of terms, and then decreased by half the product of the common difference and the number of terms minus one, results in the first term.

Example:
The total sum śreḍhīphala: the result of the progression is one hundred and five (105). There are seven terms original: sapta-pada, and the common difference original: caya is three. Tell me, what is the first term vadana: literally "face," meaning the starting value?

Numerical Setup:
First term: 0 This indicates the value to be found
Common difference: 3
Number of terms: 7
Total sum: 105
Result: The first term is 6.

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Rule for Finding the Common Difference

The total sum divided by the number of terms, minus the first term, and then divided by half the number of terms minus one, shall be the common difference original: caya.

Example:
A wise traveler first traveled eight yojanas A traditional Indian measure of distance, roughly 8-13 kilometers in a single day. Following that, by what daily increase did he travel so as to cover a total of ninety-eight yojanas in seven nights?

Numerical Setup:
First term: 8
Number of terms: 7
Total sum: 98
Result: The common difference original: uttara is 2.

Rule for Finding the Number of Terms

Progression 27 ... The text cuts off here as it prepares to define the formula for gaccha or the number of terms.