Interpretation of Bhagavad Geetha By Shri Madwacharya Pravachana: Shri Shri Shri Suvidya Theertharu, Ragi Gudda Bangalore The interpretation of Bhagavad Geetha has been made in different ways by many scholars. Shri Madvacharya has given the interpretation in “Geetha Thathparya”. Shrimadacharya has given 7 important points to be remembered while interpreting Bhagavad Geetha. 1. Lord Krishna advises Arjuna that he should fight with Kouravas as he is a Kshatriya. It means that one should perform his duty according to his varnashrama. Swadharma should be followed by everyone. 2. Kshatriya should perform Pooje (prayer) to God in the form of waging war with the enemies. Aradhane of Bhagavantha must me done through swadharma. 3. Through karma only moksha is attained. This can be attained only when one gets the knowledge of Paramathma through devotion. Paroksha jnana is got through shaastras. Aporoksha jnana is got through saakshathkaara. Vishnu Bhakti is always great. All are His dependents. invisible 4. Vishnu devotee also should do the karma. Vikarmas should be avoided. Whatever said in shrutis in the shaastraas should be followed. All the karmas should be done to please the Paramathma. 5. Jnanis also should do karma with devotion. 6. Lord Krishna advises Arjuna to fight with Kouravas. Is Arjuna jnani or ajnani? Arjuna is jnani. He is Paramathma’s aporoksha (unseen) jnani. 7. One should understand the differences among Jada, Jeeva and Paramathma. .Jada and jeeva are different. Jada and Jeeva are under His control. In all the Shaastraas it is depicted that Vishnu is Supreme and He is Sarvagunaparipurna. • All these points should be remembered always. In Bhagavad Geetha, in a shloka it is said that Jnanis need not do karma but this is applicable to only to those who have attained mukthi(liberation). Hence Krishna tells Arjuna to fight because Arjuna has not attained mukthi. All should do karma. This is the interpretation given by Shri Madvacharya. The Bhagavad Geetha starts after a period of 10 days in the battle field. When Drutharashtra did not give 5 Grammas to Pandavas, Vedavyasa tells him that he will give eyesight to Drutharashtra so that he can see how his sons will die in the war. Then Drutharahra tells that it is enough if some one narrates the happenings in the war. Vedavyasa gives the divya drushti to Sanjaya. Sanjaya after witnessing the war for 10 days, after the fall of Bhishmacharya, he narrates the picture of what happened in the battlefield. Kurukshethra means Dharmakshethra. Pandavas were dharmishtas. Duryodhana was worried about PAndavas because of Bheema. Hence he tells DronAchArya that he must protect Bhishma because his troops are lead by Bhishma. Arjuna asks Krishna to stop the chariot to find out who all have arrived at the battle field. After seeing all, Arjuna was psychologically affected and he thought that he is committing sin by fighting with them and he also had the fear of winning the battle. Hence his indriyas stopped working. Then Krishna tells Arjuna that Kouravas have done lot of adharmas, and he is not committing any sin. Krishna also tells that Kouravas are not worth for considering such things and an intelligent man like him should not think like that. AchArya’s interpretation: One should not feel sorry about living and dying. Jeevas have no death. The death is for the bodies only. Body (Deha) and Jeeva(chethana)) are different. Chethana do not die. It is inevitable that Body (Deha) has to perish one day or the other. Krishna tells Arjuna to not have too much of attachment towards body (Deha) and advises him to do his duty (karma). Acharya: Body is not permanent. Except Bhagavantha all are ashobhana. Body is nashwara(perishable). It is bhouthika (materialistic). Hence one should not feel sorry if deha(body) perishes. Varna(race), Veda and Jeeva are all nithya(permanent). Bhagavantha is the protector for all. The same Jeeva joins another jeeva after the death, but deha cannot remember its past existence. Because of previous birth’s sanskara, the same jeeva can occupy in ten dehas. When jeeva transfers from one deha to another deha, the sadhane can be increased if one has faith in GOD. All these arguments come in SAnkya. The next is Yoga. To get the knowledge of Bhagavantha, shravana and adhyayana are important factors. Adopting nishkaama karma, shravana will help in understanding GOD. Veda describes the mahime of Bhagavantha. If the thathwa in Veda is understood the moksha (liberation) is attained. Hence Krishna tells Arjuna that if he performs his karm(dutya he will attain moksha(liberation) and he can go to swarga(heaven). Veda has opined that without aspiring anything, yagna and yAga should be performed Bhagavantha says that he has created the deha and one should not expect anything from Him. Do your duty (karma) without any expectation. For this AchArya gives the explanation that this is not practicable, but one should not aspire for the bad things and bad fruits. One should perform yagna(worship) and yaaga(sacrifice) to please the GOD and should not aspire for any fruits. This can be explained like this. The medicine is taken when one gets head ache but nobody aspires for head ache to come. Veda opines that let Ajnana be there, but do your karma. Acharya has interpreted that Bhagavantha wants everyone to do the nishkAma karma. From this knowledge (jnAna) is obtained, punya and pApa will go and reaches moksha... There are two punya: vista punya and Anita punya. Loukika punya (Material virtue) will go. Rayaru is bestowing loukika punya to people who go to him by which he is becoming purer and purer and this is known as aparOksha jnAna(unseen knowledge). A Sthithapragna who knows full knowledge of GOD, all the avatharas of GOD see brightness in the day, which is darkness for others. Karma must be done by controlling the mind with the indriyas(organs). The karma should be in the form of yagna(worship or sacrifice). For fire, the roopa(quality) is prakasha(illumination). Likewise everyone should realize prakasha roopa. For sAthwiks, it is like smoke with fire. For Rajus, and for Thamus it is like child in a pregnant woman. One must understand the desire element and that should be controlled. The desire occupies indriyas(organs), mind and buddi (intelligence). These factors should be under control. The abhimAni Devathe of IndryAs is DeAbhimani. 3rd day We must find out our weakness and try to have control over it. Greediness must be avoided and should do nishkAma karma. We must try to have the blessings of IndriyAbhimani Devathes. If karma is not done then you get sin. Jagath chakra will be revolving. Jagath chakra is nothing but people. To be alive we must eat rice. Parjanya→Aditya→Surya. Rains come through Sun (Surya) Yagna is helpful in getting rains from Sun (Surya) From sathkarma only Yagna can be performed. Bhagavantha is responsible for us to perform karma also. When Veda mantras are uttered He is pleased and makes us to do karma. Vedas are compared to leaves in the trees. In the Upanishath, sathva prasaada is described. There are seven types of Anna shruthi... !) Saharan Anna: crops, grains, pashus-men. 2) Ayah: Milk, Curds. 3) Swaha: Anna (Rice) for Devathes. 4) Swaha: For Pithrus. The other three types were put in the hands of man. They are Mauna, Matha, and Deha. In other words it means Kayak, Dacha, and Manus. These three should be offered to Him. Keshena Upanishath advises that by doing karma only everyone should go through life. (Krishna and Arjuna were born in the same Manvathara, the difference being only six months). Krishna tells Arjuna that his roopa in the body is VibhUthi roopa and he has born like this from his own thinking whenever Dharma is at stake, and to establish Dharma He is born. He tells Arjuna that He will send Sathwiks people to MOksha and Thamus people to Andhathmus (hell). Acharya has interpreted this as that some sathwiks will exist in my body which is Saluda moksha; some will reach Saluki moksha; some will go to Samipya; some reach Swarupa. Why Karma should be done? From Karma Samsara begins. From Karma only, one goes to Swarga (Heaven) or Naraka (Hell). AchArya has interpreted that the fruits of karma in Heaven or Hell can not be experienced fully; a remainder of punya(virtue) or pApa(evil) will make man to come back to earth. AchArya has defined KARMA AS JEEVA. WHEN JEEVA IS DOING KARMA THAT KARMA SHOULD BE CONSIDERED AS PARAMATHMA’S KARMA. We must consider ourselves as Parathanthraru when dong karma. Life itself is Yagna. Thinking of Paramathma itself is Yagna. There are many Yagnas. But JnAna Yagna is the greatest. Arjuna asks between Sanyasa and Yoga which is greater? Sanyasa means sarvasanghaparithyagi. YOga means performance of karma. Karma must be done at any cost. Samadhi yOga(meditation) is Paramathma’s antharangha sAdhane. 4th day Aporoksha jnAna is sAdhane. To get crops first the land should be cleaned before the seeds are sowed. AchArya has told that Samadharshis also become Aporoksha Jnanis. Samadharshis are those who do not differentiate and see every one as equal. Samadharshis treat everyone according to his/her abilities(yogyathe). Bhagavantha will be present in Eka roopa in all jeevas. Bhagavantha must be seen in all species. Sama also means Paramathma with Lakshmi. Ma means Lakshmidevi. Samaadhi yOga is Antharanga Adhana(accomplishment). SamAdhi yoga is sthitha pragnathe in Bhagavantha. There are three types of Yogaadhirucharu: 1.Hiriyaru: Serving elders- Uthamaru 2.SamAnaru: Helping others, giving guidance to others – Madhyamaru. 3. Kiriyaru: Giving donations. Giving food, giving shelters - Adhamaru. Yogarudaru: Always meditating of Paramathma; serving Parents always. Krishna says that Yoga is a form of DhyAna. Krishna’s advise on SamAdhi Yoga: SamAdhi Yoga: - Isolated place should be selected. No one should be present. Body and mind should not be chanchala. (Chitha-manassu, Athma-Deha) There should not be any desire. Should not carry too many things. (Illustration: A Muni went to do samaadhi yoga in a forbidden place. There was lot of rats hindering him to do Samaadhi. Then he thought of having a cat to get rid of rats. For feeding cat with milk, he bought a cow. To look after cow he appointed a maid servant and he married her. Thus his Samaadhi ended.) SamAdhi yoga must be done near a river, and the place should be very clean. The Asana should neither be high nor low. Clean clothes should be worn before sitting for SamAdhi yOga... (While doing adhyaAnana, one should sit on dharbes or Krishnanrjana; to achieve vyrAgya one should sit on Tiger’s skin (Huli charma). One should not sit on stone, because it leads to illness. One should sit in a straight posture. One should not close the eyes completely One should concentrate between the eye brows and on the nose. The food must be taken in limited way. Too much of sleep should not be done. If possible and if energy permits SamAdhi must be done without taking food. The mind should be in the form of light burning still without air. Meditate by considering Krishna as SarvOthama. He will get asthithwa in me and he will attain Sayujya Moksha. Arjuna asks Krishna if the SamAdhi can not be continued due to disturbed mind, what happens. Krishna tells Arjuna that from constant practice and vairAgya, avoiding nishidda ahara, by detachment towards materialism, and by concentrating on Bhagavantha the mind will not get disturbed. Krishna says that even if he discontinues in the middle, he will go to heaven(swarga) and after enjoying happiness for sometime in heaven(swarga) he will again born in a rich family or yOgi family. (The example is that of Dhruva who could see Bhagavantha after doing thapas for six months only because of his sanskara in his previous birth. Dhruva was Vishnu Sharma Brahmin in his previous birth who had almost reached that stage and only six months had remained to get sAkshathkAra(manifestation or appearance) from Bhagavantha . 5th day NishkAma karma is Bahiranga(external) karma. SamAdhi yOga is Antharanga(internal)karma. Because of samskAra in the previous birth, a yOgabrashta will born in a rich family or in a yOgi’s family and he will become adyathmic(concentration of mind on Atma) later on. Bhagavantha’s knowledge (JnAna) is like diamond. Krishna preaches JnAna yOga to Arjuna. Earth, water, wind, fire, space are all under the control (adheena) of Bhagavantha. This itself is the knowledge (jnAna) about Bhagavantha. The mind decides whether to follow adyathmic life or loukika (materialistic) life. The mind wants everything. But the intellect (buddhi) has control over the mind. Krishna says that one who meditates him in mind is very great. Those who meditate my swarupa are great (Shreshta). We are very careful about ourselves. In sushupthi we will be having swarupa jnana. The egoism is ahankara thathwa. Jada prakruthi contains earth (Bhoomi) and space (Akasha). The Chithprakruthi is Lakshmi Devi who has the same energy (shakthi) as Purusha. Krishna says that Lakshmi Devi is also under his control. Krishna says that he is the creator of jagath (world). He says that he is like sutra which contains Manis. You can see Manis but cannot see sutra in a garland. Acharya interprets this as JnAnOpadesha. Kama yOgi-Manushya; jnAni yogi – Devathes; Bhakthi yOgi – Rujus, Vayu DEvaru. The water is created by Bhagavantha. The essence in the water is also created by Bhagavantha. Krishna says that one who meditates him in mind is very great. Those who meditate my swarupa(nature) are great (Shreshta). (Likewise as in Veda begins with Omkara and ends with Omkara.) Krisha says that four types of people come to him and he says that all are great (shreshtaru). 1. Rogishtaru (sufferers) 2.Aaptharu (my beloveds) 3. JignAsaru (inquisitives) 4. Wealthy people Jeevas are called adyathmic. Adhibhootha means jada padatha, jada prapancha. Adhi Daiva means the supreme in Devathes – Chathurmukha Brahma. Adhi Yagna means paramAthma inside the body. Krishna says that while traveling or at the end of the life one should think of Him only and then only he will get asthithwa(existence). How does one think of Him? One should think of Him as i) JnAni ii) Pravachana iii) buddhi preface (iv) ParipUrna (v) Osaka (vi) Brighter than one thousand Sun (vii) Inside every jeevas(soul). PourAnkika Gayathri Japa: Thath: Uuhege meerida achintya roopa Savith: Jagathige shrushtikartha Varenyam: Shreshtavaada sukha jnana ullavanu Lakshmidevigintha shreshta Bhargo: Poshaka Devasya: Kreedabhimani guna visishta Dheemahi: Anusmarana- Akhanda smruthi (khanda smruthi=100) Dhiyoyanaha Prachodayath: Nanage Buddhiyannu kodalu preraka. (volagade ninthu buddhi prerane maaduthaane) This is Apouresheya Gayathri Japa. This is Brahma Gayathri Japa. 1. Hata Yogi : Jnana is less. Moksha is there but happiness is very little.(alpa sukha) 2. Raja Yogi: Shaastra adhyana; Paramathmana chinthe. Mukthi vilamba; in moksha apaara sukha. Krisha says: I have occupied in all the things but nothing has occupied me. (Ella vasthu nannannu aashreyiside. Yaaru nannannu aashriyisilla) Shri Madwhacharya says that it means that Bhagavantha is not attached with Guna and Dharma. To reach Moksha what we should think? We must think in which roopa we are reaching Moksha. It is said that if the end comes in Uttarayana, they go near Bhagavantha. Acharya says that it has nothing to do with the kaala. He says Agni means Agni loka, Jyothi means Jyothi loka, Shukla means Abhimani Devathe, Hagalu means the direction, Uttarayana means Abhimani loka viz Bharathi loka, Praanadeva loka, Dhruva loka. In the same way if the end comes in Dakshinayana, they reach Chandra loka and come back. Acharya says that only marga or direction should be considered. Acharya also has said that it is not the kaala but in which naadi, that is whether Uttarayana naadi or Dakshinayana naadi the end has come should be considered. Arjuna asks Krishna how one should pray him? Krishna says it is very simple and His mahatme is very large and He is easily reachable by every one. “Pathram, Pushpam, Phalam, Thoyam” Pushpam means hoovu(flower), Phalam means hannu(fruit), Thoyam means neeru(water). Paramathma says that Bhakti (devotion) is most important. Paramathma says that He will see the devotion in the heart of persons who pray him. The internal meanings of “Patram, Pushpam, Phalam, Thoyam”. Pathram means Veda mantra, Gayathri japa. Jagath is Vruksha. The trunk (Buda) is Paramathma. Krishna says that the leaves are Vedas. If Veda Mantras, Gayathri japa, Suladis are done that itself is Pathram. If the Vedas are destroyed then Jagath will vanish. Pushpam – flowers – means poshaka and paalika. That is protector. Adyathmika pushpa is the protector for the whole world(Jagath). There are eight different bhava pushpas which never dries and are invaluable. 1. Ahimsa pushpa: Not to torture body and mind. 2. Indriya pushpa: Controlling of Indriyas 3. Sarva Bhutha daya: - Love towards all animals. 4. Sathya pushpa: Only sweet things should be conveyed to good people. 5. Jnana pushpa: Thathwa Jnana of Bhagavantha. 6. Thapas pushpa: Aalochana, Chinthane. (Meditation?) 7. Kriya pushpa: Dharma 8. Dhyana: Samadhi yoga. The above eight Bhava pushpas are the controllers of the Jagath. Thoyam: Offering water to Bhagavantha . Krishna says whatever you do, offer it to me and I receive everything. Krishna says if one prays me with full of devotion, and even if that person commits sin he is Sadhu to me. Acharya says that it is not applicable to us. It is applicable to Devathes only. To be clearer it is applicable to Indra, Chandra, and Sugreeva. Compiled by: - J. Sudhindra Simha, Bangalore (Bangalore) .

I am sending another version of "Interpolation" as the previous one was not in order. Interpolation “Interpolation consists in reading a value which lies between two extreme points, and Extrapolation means reading a value that lies outside the two.” – W.M. Herper We use statistical technique known as Interpolation or extrapolation when we have to estimate a value which is not known in a given series. Interpolation refers to the insertion of value in-between the given series of values and extrapolation refers to projecting a value for the future. Hence interpolation helps us to get the missing link in the series, and extrapolation helps us to forecast. In simple interpolation, it is assumed that there is a linear relationship between x and y, where x being the independent variable and y the dependent variable. Suppose we are given the values x0, x1, x2, xn of x and let the corresponding values of y be y0, y1, y2, yn respectively. If we want to estimate the value of yx for any value of x between the limits x0 and xn, this can be done by applying interpolation technique. All the formulae of interpolation are based on the fundamental assumption that the given data can be expressed as a polynomial function with a fair degree of accuracy. (Polynomial functions are functions that have this form: f(x) = anxn + an-1xn-1 + ... + a1x + a0 The value of n must be a nonnegative integer. That is, it must be whole number; it is equal to zero or a positive integer. Positive integers are all the whole numbers greater than zero: 1, 2, 3, 4, 5, Negative integers are all the opposites of these whole numbers: -1, -2, -3, -4, -5, We do not consider zero to be a positive or negative number. The coefficients, as they are called, are an, an-1, a1, a0. These are real numbers. Polynomial functions are functions that have this form: f(x) = anxn + an-1xn-1 + ... + a1x + a0 The value of n must be a nonnegative integer. That is, it must be whole number; it is equal to zero or a positive integer. The coefficients, as they are called, are an, an-1, a1, a0. These are real numbers.) Finite Differences and Operators In problems of interpolation, the independent variable x in the functional relation y=f(x) is called the argument, and the dependent variable y, the entry. Let x0, x1, x2,…xn-1, xn be a set of equidistant values of the argument x, i.e. x1-x0=x2-x1=……=xn-xn-1 = h (constant) and let y0, y1, y2, …….., yn-1, yn be the corresponding values of y. Then y1-y0, y2-y1, ……, yn-yn-1 are called the fist differences (or finite differences of first order) of the function y. Denoting these differences by ∆y0 = y1-y0, ∆y1 = y2-y1…... ∆yn-1 = yn – yn-1 The differences of these first differences, i.e. the differences ∆y1 - ∆y0, ∆y2 - ∆y1, ∆yn-1 - ∆yn-2 are called the second differences. Denoting them by ∆2y0, ∆2y1, etc., we have ∆2y0 = ∆y1 - ∆y0, ∆2y1 = ∆y2 - ∆y1,….., ∆2yn-2 = ∆yn-1 - ∆yn-2 Similarly, we can define the third, fourth and higher order of differences. Newton’s Forward Interpolation Formula Let x0, x1, x2,…xn-1, xn be a set of equidistant values of the argument (or independent variable) x i.e. x1-x0=x2-x1=……=xn-xn-1 = h (say) and y0, y1, y2, …….., yn-1, yn be the corresponding values of the function y=f(x). Then the value of y corresponding to a specified value of x lying near the beginning of the tabulated values is given by the Newton’s forward interpolation formula: y = y0 + (u/1) y0 + {u (u-1)/1.2}2y0 + {u (u-1) (u-2)/1.2.3} 3y0 + ------------- --------------+ {u (u-1) u-2)………….. (u-n+1)/1.2.3…..n} ny0, where u = (x-x0/h. Newton’s Backward Interpolation Formula Let x0, x1, x2,...xn-1, xn be a set of equidistant values of the argument (or independent variable) x and y0, y1, y2, …….., yn-1, yn be the corresponding values of the function y=f(x). Then the value of y corresponding to a specified value of x lying near the end of the tabulated values is given by the Newton’s backward interpolation formula: y = yn + (u/1) yn-1 + {u(u+1)/1.2}2yn-2 + {u(u+1)(u+2)/1.2.3}3yn-3 + --------------------------- + {u(u+1)u+2)…………..(u+n-1)/1.2.3…..n}ny0, where u = (x-x0/h and h = x1-x0=x2-x1=……=xn-xn-1 Lagrange’s Interpolation formula Let x0, x1, x2, ….xn-1, xn be a set of values of the argument (or independent variable) x (not necessarily equidistant) and y0, y1, y2, …….., yn-1, yn be the corresponding values of the function y=f(x). Then the value of y corresponding to a specified value of x lying between the tabulated values is given by the Lagrange’s Interpolation formula: y = {(x-x1) (x-x2)…. (x-xn)/(x0-x1) (x0-x2)…..(x1-xn)}*yo + {(x-x0) (x-x2)…. (x-xn)/(x1-x0) (x1-x2)…. (x1-xn)}*y1 + ……………………………………………………….. + {(x-x0) (x-x1)…. (x-xn-1)/ (xn-x0) (xn-x1)…. (xn-xn-1)}*yn Inverse Interpolation Lagrange’s Interpolation formula may also be used to find the value of argument x for a given value of the function y. This process is known as Inverse Interpolation. Lagrange’s formula for inverse interpolation is given by X = {(y-y1) (y-y2) …. (y-yn)}/ (y0-y1) (y0-y2)…. (y0-yn)}*x0 + {(y-y0)(y-y2) …. (y-yn)/ (y1-y0) (y1-y2)….(y1-yn)}*x1 + {(y-y0) (y-y1)…. (y-yn-1)/ (yn-y0) (yn-y1)….(yn-yn-1)}*xn INTERPOLATION & EXTRAPOLATION 1. Given the following logarithms, interpolate the value of log 20.5 x 20.4 20.5 20.6 20.7 Log x 1.3096 - 1.3139 1.3160 Here the arguments x0 = 20.4, x1 = 20.5, x2 = 20.6 and x3 = 20.7 are equidistant. y1 is not known (missing). This unknown entry y1 can be interpolated by binomial method. x 20.4 x0 20.5 x1 20.6 x2 20.7 x3 Y = log x 1.3096 y0 - y1 1.3139 y2 1.3160 y3 There are n = 3 known entries. The entry y1 is missing. Therefore ∆3y0 = 0. Thus by using Pascal’s triangle ∆3y0 = 0. Y3 – 3y2 +3y1 – y0 = 0 1.3160 – 3*1.3139 + 3 y1 – 1.3096 = 0 3 y1 = 1.3096 – 1.3160 + 3*1.313 3 y1 = 3.9353 y1 = 1.3118 2. The following are the annual premiums for an insurance policy of Rs. 1000. Calculate the annual premium for a person aged 25 years. Age of persons(years) 20 25 30 36 Premium(rupees) 23 - 30 35 Age of persons 20 x0 25 x1 30 x2 35 x3 Premium (rupees) 23 y0 - y1 30 y2 35 y3 There are n = 3 known entries. The entry y1 is missing. Therefore ∆3y0 = 0. Thus by using Pascal’s triangle ∆3y0 = 0. y3 – 3y2 +3y1 – y0 = 0 35 – 3*30 + 3* y1 – 23 = 0 3* y1 = 90 +23 – 35 3* y1 = 78 or y1 = 26 Therefore, annual premium for a person aged 25 years is y1 =Rs. 26. 3. Find the missing figure in the following series by binomial method x 1 2 3 4 5 6 y 200 - 240 255 260 265 x 1 2 3 4 5 6 y 200 y0 - y1 240 y2 255 y3 260 y4 265 y5 Here n = 5 Here, the arguments are equidistant and one entry, namely y1 is missing. Using Pascal’s triangle ∆5y0 = 0. y5 - 5y4 + 10y3 - 10y2 + 5y1 - y0 = 0 5y1 = yo + 10y2 – 10y3 + 5y4 – y5 5y1 = 200 +10*240 – 10*240 + 5*260 – 265 5y1 = 1085 y1 = 217 4. The following data gives the sales of a manufacturing firm in six successive years. Using the data estimate the sales in 2013’ Year 2007 2008 2009 2010 2011 2012 Sales (Thousand units) 120 140 160 180 200 22 Year 2007 2008 2009 2010 2011 2012 Sales (Thousand units) 120 y0 140 y1 160 y2 180 y3 200 y4 225 y5 Here, the arguments are equidistant But none of the enties is missing. The range of the arguments is (2007, 2011). We have to estimate the sales in 2013. The value 2013 is outside the range. Therefore, we are using binomial method for extrapolation. For 2013, y6 is the sales value. There are n = 6 known entries. Therefore ∆6y0 = 0. Thus by using Pascal’s triangle, ∆6y0 = 0 y6 - 6y5 + 15y4 - 20y3 + 15y2 - 6y1 + y0 = 0 y6 – 6*225 + 15*200 – 20*180 + 15*160 – 6*140 + 120 y6 = 6*225 – 15*200 +20*180 – 15*160 + 6*140 – 120 y6 = 270 (Thousand units) 5. Interpolate the missing value x 180 190 200 210 220 y 87 83 79 - 71 x 180 190 200 210 220 y 87 y0 83 y1 79 y2 - y3 71 y4 Here n = 4 known entries. The entry y3 is missing. Therefore, using Pascal’s triangle ∆4y 0= 0. Y4 – 4y3 + 6y2 – 4y1 + y0 = 0 71 - 4y3 + 6*79 – 4*83 + 87 = 0 4y3 = 71 + 6*79 – 4*83 + 87 4y3 = 300 y3 = 75 6. If e02 = 1.2214 and e04 = 1.4918, interpolate the value of e03 by binomial method. e02 e03 e04 1.2214 y0 - y1 1.4918 y2 Here the known entries n = 2 Thus by applying Pascal’s triangle ∆2y0 = 0 y2 – 2y1 + y0 = 0 1.4918 - 2y1 + 1.2214 = 0 2y1 = 2.7132 y1 = 1.3566 7. From the following data, estimate the population of India in census year 1981. Census year 1991 2001 2011 Population(crore) 85 103 121 Census year 1981 1991 2001 2011 Population(crore) - y0 85 y1 103 y2 121 y3 Here, the arguments are equidistant But none of the enties is missing. The range of the arguments is (1991, 2011). We have to estimate the population in 1981. The value 1981 is outside the range. Therefore, we are using binomial method for extrapolation. For 1981, y0 is the population value. There are n = 3 known entries. The entry y1 is missing. Therefore ∆3y0 = 0. Thus by using Pascal’s triangle ∆3y0 = 0. y3 – 3y2 +3y1 – y0 = 0 121 – 3*103+ 3*85 – y0 = 0 y0 = 121 – 309 + 255 y0 = 67 Hence the population in 1981 is 67 crores 8. From the missing entry in the following data regarding average yield of coconut trees. Age(years) 1 2 3 4 5 Average yield 138 151 193 - 296 Age(years) 1 2 3 4 5 Average yield y 138 y0 151 Y1 193 y2 - y3 296 y4 Here n = 4 known entries. The entry y3 is missing. Therefore, using Pascal’s triangle ∆4y 0= 0. Y4 – 4y3 + 6y2 – 4y1 + y0 = 0 296 - 4y3 + 6*193 – 4*151 + 138 = 0 4y3 = 296 + 1158 – 604 + 138 4y3 = 988 y3 = 247