## 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 \ \$ $\ds d$ $=$ $\ds \dfrac {66} {21}$ $\ds$ $=$ $\ds \dfrac {22} 7$

$\blacksquare$