Geometrical Representation of Arithmetic Series – In the Classroom

Proposed Lesson Plan

Objectives

• To show a simple yet elegant relation between arithmetic and geometry.
• To read and interpret an historical mathematical text in translation.
• To arouse historical and cultural curiosity.

Required Background

• Secondary school geometry, arithmetic, and algebra.
• Capacity to manipulate simple equations.

Activities

• Brief background of arithmetic in ancient Indian texts (5 minutes).
• Recall basics of arithmetic progression (5 minutes).
• Description of sum of series expressed geometrically (10 minutes). The nuances may be included or not.
• Reading of translated verses (10 minutes).
• Examples with drawings (10-15 minutes).

Assessment: Problems

The following exercises may be given as part of a written assignment or an examination. All but Problems 1 and 3 are from Sanskrit texts as indicated.

1. Prove that the formula for the sum of an arithmetic series of \(n\) terms with given first term \(a\) and constant difference \(d\) is correct using modern notation and methods.
2. On an expedition to seize his enemy's elephants, a king marched two units the first day.  With what increasing rate of daily march did he proceed, if he reached his enemy's city, a distance of eighty units, in a week? (Līlāvatī, v. 124) Draw the associated series figure.
3. Draw a series figure to represent the sum of the first 10 positive odd integers. Obtain a general formula by replacing 10 by \(n\) and show that this agrees with the area of the corresponding series-figure.
4. (This problem could be considered as an amusing variation of the last.) In a dice game, two people alternately win 4, 3, 2, and 2 casts of dice from each other. The stake-money begins with 1 and increases by 2 with each successive roll. (Pāṭīgaṇita, v. 115) Draw the corresponding series figure (you may use different colors for the opponents) and decide who the winner is.
5. One man travels with an initial speed of 3 yojanas per day and acceleration 1 yojana per day. Another man travels with the (constant) speed of 10 yojanas per day. In what time will they cover the same distance? (Pāṭīgaṇita, v. 111) Draw the corresponding series-figures. (The yojana was a measure of distance very roughly between 5 and 15 kilometers (roughly, 3 to 9 miles).)
6. If the first term of a series is 3, the common difference 7, and the number of terms \(\frac1{7},\) what is the area of its series figure? (Gaṇitakaumudī, Vol. 2, v. 63) Can you draw a trapezium corresponding to this series?
7. Find the area of the series figure where the first term of a series is \(\frac1{2},\) the common difference 3, and the number of terms \(3\frac{1}{3}.\) (Gaṇitakaumudī, Vol 2, v. 62.2) Draw the figure.
8. A man gave his son-in-law sixteen units of money the first day; and diminished the sum by two units per day. How many units had he bestowed when the ninth day was past? (Brāhmasphuṭasiddhānta of Brahmagupta, Chapter 12) Draw the series-figure.
9. (Challenge problem) What is the number of cows at the end of 20 years, starting with one cow that gives birth to one calf every year and every calf in turn beginning to reproduce at the age of 3 years? (Gaṇitakaumudī, Vol. 1, p. 126)

Cryptic Answers

1. A constructive derivation and/or a proof by mathematical induction may be given.
2. \(d= \frac{22}{7}.\)
3. \(S_{10}=100, S_n=n^2,\) triangle.
4. First  \(1+3+5+7,\) second \(9+11+13,\) etc.; second player wins; triangle with trapeziums inside.
5. \(15\) days, rectangle and trapezium.
6. \(A =0,\) upper and lower triangles of equal area.
7. \(A=\frac{40}{3},\) base negative, face positive.
8. \(S_9=72,\) base positive, face negative.
9. \(2745\) if each cow produces her first calf just before or just as she turns 3 years old; \(907\) if each cow produces her first calf only after she turns 3 years old and before she turns 4 years old.