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hartnn Group Title

TUTORIAL on Arithmetic Progression (AP or Arithmetic Sequence), Geometric Progression (GP), Harmonic Progression(HP). This was one of my favorite topics in junior college. P.S.: This is my first tutorial. So don’t be hard on me :P

  • 2 years ago
  • 2 years ago

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  1. hartnn Group Title
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    \[\large{\color{Blue}{T}\color{Yellow}{U}\color{Magenta}{T}\color{Pink}{O}\color{MidnightBlue}{R} I\color{Blue}{A}\color{Green}{L}!}\] \[\color{Red}{Arithmetic Progression (AP):}\] A sequence is in arithmetic progression when its next term to previous term ‘difference’ is always constant. As an example, if I say a frog jumps every 2 seconds starting right now which we take as t=0 sec, the next jump will be on t=2 sec. When will the next jump be? Yes, you are correct, it will be at t=4 sec. Now how did u get 4 ? Let me tell you, you added the difference d=2 to previous term (2). So to get next term of AP you always add the common difference to current term, and to get previous term you always subtract the common difference from current term. A more general formula to get the ‘n’ th term of an AP is: \[T_n=a_1+(n-1)d\] Where a_1is the first term and d is the common difference of the AP. So the general terms of AP are \[ a_1, a_1+d,a_1+2d,a_1+3d,…..a_1+(n-1)d.\] When needed to find the sum of ‘n’ terms of an AP, you use: \[S_n=\frac{n}{2}(2a_1+(n-1)d)\] But you have a shortcut formula, when you have the 1st and the last term of AP: \[S_n=\frac{(a_1+a_n)}{2}\] ,where an is the last term. The sum of infinite Arithmetic Sequence is essentially infinite. But, if a=b=0,the sum is 0. The product of ‘n’ terms of a finite AP is \[d^n \frac{\Gamma(a_1/d+n)}{\Gamma(a_1/d)}\] when a1/d is positive integer. Note: Ignore this if you are unfamiliar with Gamma Functions. \[\color{Red}{Geometric Progression (GP):}\] A sequence is in arithmetic progression when its next term to previous term ‘ratio’ is always constant. Classical example of Geometric progression is population growth. If I say population doubles every 15 years, what does it mean? If the population is 7 billion in 2012, the population in 2027 will be 14 billion. So what will be the population in 2042? Yes! you are again correct it will be 28 billion. How did u get 28? You multiplied 14 by common ratio 2 to get 28. So to get next term of GP you always multiply the common ratio to current term, and to get previous term you always divide the current term by common ratio. A more general formula to get the ‘n’ th term of a GP is: \[T_n=a_1r^{n-1}\] Where a1 is the first term and r is the common ratio of the GP. So the general terms of GP are \[a_1,a_1r,a_1r^2,…..a_1r^{n-1}.\] When needed to find the sum of ‘n’ terms of a GP, you use: \[S_n=a_1\frac{(1-r^n)}{(1-r)}\] when |r|<1 \[S_n=a_1\frac{(r^n-1)}{(r-1)}\] when |r|>1 The sum of infinite geometric sequence with |r|>1 , is essentially infinite. The sum of infinite geometric sequence with |r|<1 , is \[S_ \infty=\frac{a}{(1-r)}\]. The product of ‘n’ terms of a finite GP is \[P=(a_1a_{(n+1)})^{\frac{(n+1)}{2}}\] \[\color{Red}{Harmonic Progression (HP):}\] I would call HP as reciprocal of AP, because if we take the reciprocal of the terms in AP, the new terms would be in HP or equivalently, if we take the reciprocal of the terms in HP, the new terms would be in AP. The general terms of HP are : \[a_1,\frac{a_1}{(1+d)},\frac{a_1}{(1+2d)}….\frac{a_1}{(1+(n-1)d)} \] There is no accurate simple general formula for sum to ‘n’ terms of the series. ______________________________________________________________________ Now, lets play with only 3 term sequences. If a,b,c are in AP, then b is the arithmetic mean(AM) of a and c and is given by \[b=\frac{(a+c)}{2}\] This you get from the fact that the difference between consecutive terms are equal,so b-a=c-b or a+c=2b or b=(a+c)/2 If A,B,C are in GP, then B is the geometric mean(GM) of A and C and is given by \[B=\sqrt{AC}\] This you get from the fact that the ratio between consecutive terms are equal,so A/B=B/C,so AC=B^2 So B=sqrt(AC). If X,Y,Z are in HP, then Y is the harmonic mean(HM) between X and Z given by \[Y=\frac{2XZ}{(X+Z)}\] Also, this relation is very useful : The geometric mean of 2 numbers is the geometric mean of arithmetic mean and harmonic mean of those two numbers. Meaning: \[GM^2=AM.HM\] Or here \[B^2=b.Y\](if a=A=X,c=C=Z)

    • 2 years ago
  2. mukushla Group Title
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    u have great skill in Progressions...:)

    • 2 years ago
  3. hartnn Group Title
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    thanks :) as i told it was my favorite topic,so i practiced a lot...

    • 2 years ago
  4. Yahoo! Group Title
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    Can...u Do AGP @hartnn

    • 2 years ago
  5. hartnn Group Title
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    my initial intention was to include AGP also and typical examples of it,but this tutorial was already long enough!

    • 2 years ago
  6. experimentX Group Title
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    you guys have patience

    • 2 years ago
  7. lgbasallote Group Title
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    what's up with those kinds of latex lol

    • 2 years ago
  8. sauravshakya Group Title
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    great work........

    • 2 years ago
  9. sauravshakya Group Title
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    @Abhishekjha PLZ SEE THIS

    • 2 years ago
  10. EulerGroupie Group Title
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    Man! I have trouble with this site sometimes. I can't seem to message people and I can't read the LaTeX on this posting.

    • 2 years ago
  11. hartnn Group Title
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    Maybe,because of slow net speed....

    • 2 years ago
  12. EulerGroupie Group Title
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    I have pretty good gear and connection speed, but I frequently seem to have compatability issues. I turned off pop up blocker, but still no messages out. I love this site, so I will keep investigating. If I get to the point where I can read it, I would be happy to thoroughly check this posting out. Thank you for pointing it out.

    • 2 years ago
  13. hartnn Group Title
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    ok,thanks for your time :)

    • 2 years ago
  14. mathslover Group Title
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    Nice turorial hartnn

    • 2 years ago
  15. waterineyes Group Title
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    Use \; , \: , \quad or \qquad to give spaces between two words..

    • 2 years ago
  16. waterineyes Group Title
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    Actually, AP is my favorite topic.. Great work @hartnn .. This will help a lot..

    • 2 years ago
  17. hartnn Group Title
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    thanks @waterineyes i used {space} for spaces,though not here...

    • 2 years ago
  18. ajprincess Group Title
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    Great work @hartnn

    • 2 years ago
  19. waterineyes Group Title
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    And don't think I am trying to hard on you.. Ha ha ha...

    • 2 years ago
  20. hartnn Group Title
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    even i don't think that :) thanks for wonderful replies, @waterineyes and @ajprincess

    • 2 years ago
  21. waterineyes Group Title
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    Oh @hartnn sorry to say, but you are missing \(\color{blue}{n}\) in Sum formula for AP.. The Second sum formula that is related to fist and last terms..

    • 2 years ago
  22. waterineyes Group Title
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    *first..

    • 2 years ago
  23. hartnn Group Title
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    oh yes! thanks for pointing it out. \[S_n=n\frac{(a_1+a_n)}{2} \]

    • 2 years ago
  24. hartnn Group Title
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    i don't think i can edit that now.....can i?

    • 2 years ago
  25. waterineyes Group Title
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    Yes you can..

    • 2 years ago
  26. waterineyes Group Title
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    Copy the whole text and go to "Ask a Question" box and paste it their.. Ha ha ha ha...

    • 2 years ago
  27. waterineyes Group Title
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    Oh God my english.. *there..

    • 2 years ago
  28. hartnn Group Title
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    yeah,i can always do that :P but m gonna ask a new question.....

    • 2 years ago
  29. waterineyes Group Title
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    Yeah sure why not..

    • 2 years ago
  30. hartnn Group Title
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    any suggestions for next tutorial? i am thinking about: 1. Definite integration. 2.Partial Fractions. 3.Limits. 4.Solving 3rd degree Equation(1 variable) Which ONE should i give next? or any other topic whose questions asked by many...??

    • 2 years ago
  31. waterineyes Group Title
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    In my opinion, Go for Limits or Definite Integrals.. Limits being the first preference..

    • 2 years ago
  32. mukushla Group Title
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    and also 4...including rational root theorem

    • 2 years ago
  33. hartnn Group Title
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    ok,i'll first write on solving cubic eq as it will be short,then next week on limits as it will be long.

    • 2 years ago
  34. waterineyes Group Title
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    Sure..

    • 2 years ago
  35. sami-21 Group Title
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    Excellent , i must say you are Oracle in AP :)

    • 2 years ago
  36. hartnn Group Title
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    thanks :)

    • 2 years ago
  37. kimii Group Title
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    this is good ..

    • one year ago
  38. hartnn Group Title
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    thanks :)

    • one year ago
  39. kimii Group Title
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    thi is so helpful . i hope i will answer my quiz for monday on school .

    • one year ago
  40. kryton1212 Group Title
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    wow, amazing. It seems helpful:)

    • one year ago
  41. hartnn Group Title
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    thanks :)

    • one year ago
  42. Princy123 Group Title
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    thnx. it is so helpful... @hartnn

    • 11 months ago
  43. hartnn Group Title
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    welcome ^_^

    • 11 months ago
  44. Compassionate Group Title
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    I like to post on dead threads.

    • 10 months ago
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