# Bhaskara II Acharya/Lilavati/Chapter V/124

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## Bhaskara II Acharya:

## 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?*

## Solution

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

## Proof

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 $T$.

Then:

\(\ds T\) | \(=\) | \(\ds n \paren {a_0 + \dfrac {n - 1} 2 d}\) | Sum of Arithmetic Sequence | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds 80\) | \(=\) | \(\ds 7 \paren {2 + \dfrac {7 - 1} 2 d}\) | plugging in the numbers | ||||||||||

\(\ds \) | \(=\) | \(\ds 14 + 21 d\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds d\) | \(=\) | \(\ds \dfrac {66} {21}\) | |||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {22} 7\) |

$\blacksquare$

## Sources

- 1816: John Taylor:
*Lilawati: or A Treatise on Arithmetic and Geometry by Bhascara Acharya*: Part $\text {I}$: Arithmetic: Chap. $\text {IV}$: Of Progressions - 1817: Henry Thomas Colebrooke:
*Algebra, with Arithmetic and Mensuration*: Arithmetic (Lilavati): Chapter $\text {V}$: Progressions: $124$ - 1893: Haran Chandra Banerji:
*Colebrooke's Translation of the Lilavati*: Chapter $\text {V}$: Progressions: $124$ - 1965: Henrietta Midonick:
*The Treasury of Mathematics: Volume $\text { 1 }$* - 1976: Howard Eves:
*Introduction to the History of Mathematics*(4th ed.) - 1992: David Wells:
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