Arithmetic, geometric and harmonic progression

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Arithmetic, geometric and harmonic progression

  1. 1. ARITHMETIC, GEOMETRIC AND HARMONIC PROGRESSION by : DR. T.K. JAIN AFTERSCHO ☺ OL centre for social entrepreneurship sivakamu veterinary hospital road bikaner 334001 rajasthan, india FOR – PGPSE / CSE PARTICIPANTS [email_address] mobile : 91+9414430763
  2. 2. My words..... My purpose here is to give an introduction of arithmetic, geometric and harmonic progression . I welcome your suggestions. I also request you to help me in spreading social entrepreneurship across the globe – for which I need support of you people – not of any VIP. With your help, I can spread the ideas – for which we stand....
  3. 3. What is ARITHMETIC PROGRESSION ? 3 tests of AP : 1 there must be a fixed interval between any two digits in a seris 2. there must not be any other reason for the numbers to come in that order 3 the numbers either increase or decrease
  4. 4. Example of AP ? 8,13,18, 23,28,33 .... Here we have an AP, because there is a fixed gap and the series is increasing in a series we have 3 terms : a = first term = 8 d = difference between any two terms = 5 n = number of terms
  5. 5. Basic formula of AP ? In order to find sum of some terms, you can use any of these formula : 1. n/2 ( 2a+(n-1)d) 2 n/2 (first term + last term)
  6. 6. Example of the formula : What will be the sum of first 100 terms of this series : 8,13,18,23,28 ... n/2 ( 2a+(n-1)d) n = 100, a = 8, d = 5 100/2 (2*8 + (100-1)*5) =50(16 + 495) =25550 answer
  7. 7. How to identify a term in an AP ? In order to identify any term, use this formula : Tn = a + (n-1)d what is 30 th term in this series : 8,13,18,23,28 ... = 8+(30-1)5 =8+145 =153 ans
  8. 8. What wil be 17 th term in the series : 103, 123, 143, 163 .... ? Formula tn= a + (n-1) d where, a = starting term = 103 d = difference (123- 103 = 20) , n = number of term =103 + (17-1) * 20 =103 + 320 = 423 answer
  9. 9. There are 31 terms in an AP . The first term is 100340 and the last term is 100580. What is the sum of that series ? Formula =n/2 * (first term + last term) n = 31, first term = 100340, last term =100580 =31/2 *(100340+100580) =3114260 answer
  10. 10. What is GP ? (geometric progression) When a series is rising at an increasing fast speed, it is geometric progression example : 2,4,8,16,32,64 .....here a= 2 (starting term) raio = 16/8 = 2 (you make pick up any two terms, the ratio will be the same)
  11. 11. Essential components of GP? Take any GP : Let us say : 3,9,27, 81,243 There are 3 terms : 1. starting term = a = 3 2. growth as ratio of two terms : r = 27/9 =3 3.n = number of terms = n=5
  12. 12. Can ratio be less than 1 ? Yes, in that case the series will be a decreasing series example : 81,27,9,3,1,1/3, 1/9,1/27 ....
  13. 13. What is the sum of this series for 5 terms: 1^2, 2^2, 3^2 . . . . Formula :( n (n+1) (2n+1) )/ 6 =(5(6)(11))/6 =55 answer
  14. 14. What is the sum of this series for 5 terms: 1^3, 2^3, 3^3 . . . . Formula = ((n(n+1)/2)^2) (5(5+1)/2)^2 = 225 answer
  15. 15. What is the sum of this infinite series : 512, 256,128,64......for first 10 terms ??? Formula = a / (1-r) a=512, r = 256/512 = ½ = .5 = 512/ (1-.5) = 1024 ans.
  16. 16. What is sum of this series : 2,4,8,16 .for first 30 terms ??? 'Sn = a (r^n -1) / (r-1) a= starting term r = ratio n = 30 =2(2^30 – 1) / (2-1) =2 * 1073741823 =2147483646 answer
  17. 17. What is the sum of first 8 terms of this series : 8, 64, 512 ..... This is GP formula for sum is : a (r^n – 1) / (r-1) a = first term (8), r = ratio = 64/8 = 8 8 * (8^8 - 1) / (8-1) =19173960 answer
  18. 18. What is the sum of first 5 terms of this series : 81,27,9 ... This is GP formula for sum = a (1-r^n) / (1-r) a = 81, r =27/81 = 1/3, n =5 = 81 (1-(1/3)^5) / (1-(1/3)) =121 answer
  19. 19. Which of these is geomtric progression ? 3,5,88,122 3,5,7,9,11... 4,16,256 ... only the last one (4,16,256....) is a GP
  20. 20. In order to study at ISB Hyderabad, Mayank starts saving money. First day he saves Rs. 200000. Next day he saves half of this money. This process continues. Now this becomes his habit. If Mayank continues saving like this, how much money can he save in his lifetime? Formula = a / (1-r) a = first term (2,00,000) , r = ratio (½) =200000/.5 = 4 lakhs only answer
  21. 21. If after 1 year, Mayank can get Rs. 50 lakhs in one shot. How much is it worth now. Rate of interest is 20% ? Formula of present value =( amount*100) / (100 + (rate * time) ) =(5000000*100)/(120) =4166666 answer
  22. 22. What is the worth of Rs. 100000 now after 10 years at 20% annual compounding interest ? Formula = amount (1+rate ) ^ n n = number of years 100000 * (1+.2)^10 =619174 answer
  23. 23. A goes to his school @ 20 km per hour and returns back to home @ 30 km per hour. What is average speed ? Average of 20 and 30 is 25, but here we have to calculate harmonic mean. Formula of harmonic mean =reverse of 1/n ( 1/x + 1/y) or : (2XY / (X+Y)) =1/2 (1/20 + 1/30) = ½ * 50 / 600 = 24 answer
  24. 24. What is harmonic progression ? It is just reverse of AP for example : 2,4,6,8 is an AP HP = 1/2, 1/4, 1/6 ,1/8 harmonic progression will convert into AP when you reverse it.
  25. 25. Find AM,GM,HM between 20 and 30 ? AM= (20+30)/2 = 25 GM = sqrt (20*30) = 24.49 HM = REVERSE OF ½ ( 1/20+1/30)=24
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