Bhaskara II Acharya/Lilavati/Chapter V/124

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: Curious and Interesting Puzzles ... (previous) ... (next): Indian Puzzles: $55$