Bhaskara II Acharya/Lilavati/Chapter V/124

From ProofWiki
Jump to navigation Jump to search

Bhaskara II Acharya: Lilavati: Chapter $\text {V}$: Progressions: $124$

In an expedition to seize his enemy's elephants, a king marched $2$ yojanas on the first day.
Say, intelligent calculator, with what increasing rate of daily march did he proceed, since he reached his foe's city, a distance of $80$ yojanas, in a week?


His march was increased by the rate of $\dfrac {22} 7$ yojanas per day.


We are given that the daily rate of march increases each day by an arithmetic sequence.

Let the initial term be denoted by $a_0$.

Let the common difference, which we are to find, be denoted by $d$.

Let the number of days march be $n$.

Let the total number of yojanas marched be $T$.


\(\ds T\) \(=\) \(\ds n \paren {a_0 + \dfrac {n - 1} 2 d}\) Sum of Arithmetic Sequence
\(\ds \leadsto \ \ \) \(\ds 80\) \(=\) \(\ds 7 \paren {2 + \dfrac {7 - 1} 2 d}\) plugging in the numbers
\(\ds \) \(=\) \(\ds 14 + 21 d\)
\(\ds \leadsto \ \ \) \(\ds d\) \(=\) \(\ds \dfrac {66} {21}\)
\(\ds \) \(=\) \(\ds \dfrac {22} 7\)