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Karon86

  • one year ago

What is the 100th term of sequence: 1,3,6,10,15,.... Write a explicit formula

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  1. jim_thompson5910
    • one year ago
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    |dw:1443915861353:dw|

  2. jim_thompson5910
    • one year ago
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    the jump from 1 to 3 is +2 |dw:1443915905024:dw|

  3. jim_thompson5910
    • one year ago
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    the jump from 3 to 6 is +3 |dw:1443915932403:dw|

  4. jim_thompson5910
    • one year ago
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    the jump from 6 to 10 is +4 |dw:1443915954079:dw|

  5. jim_thompson5910
    • one year ago
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    the jump from 10 to 15 is +5 |dw:1443915976631:dw|

  6. jim_thompson5910
    • one year ago
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    so what you can do is keep this pattern up until you have generated 100 terms OR you can find an explicit closed formula so you can avoid using recursion (recursion is long and tedious for a lot of terms)

  7. Karon86
    • one year ago
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    I want an explicit formula

  8. jim_thompson5910
    • one year ago
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    I prefer the second method each difference (+2, +3, +4, +5, ...) is separated by 1 each time this second level of finite differences suggests we have a 2nd degree polynomial

  9. jim_thompson5910
    • one year ago
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    |dw:1443916308731:dw|

  10. jim_thompson5910
    • one year ago
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    the general quadratic is ax^2 + bx + c

  11. jim_thompson5910
    • one year ago
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    plug in x = 1 and we get ax^2 + bx + c = a*1^2 + b*1 + c = 1a+1b+c = a+b+c the input is 1, and the output is 1 |dw:1443916381381:dw| so a+b+c = 1

  12. jim_thompson5910
    • one year ago
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    next plug in x = 2 (circled) to get ax^2 + bx + c a*2^2 + b*2 + c 4a+2b+c this is equal to the output of 3 (circled) in the second row |dw:1443916438115:dw| so 4a+2b+c = 3

  13. jim_thompson5910
    • one year ago
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    |dw:1443916456906:dw| plug in x = 3 ax^2 + bx + c a*3^2 + b*3 + c 9a+3b+c the output is 6, so 9a+3b+c = 6

  14. jim_thompson5910
    • one year ago
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    we now have this system of equations a+b+c = 1 4a+2b+c = 3 9a+3b+c = 6

  15. Karon86
    • one year ago
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    I am still here

  16. jim_thompson5910
    • one year ago
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    do you know how to solve for a,b,c?

  17. jim_thompson5910
    • one year ago
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    you have a number of options 1) elimination 2) substitution 3) matrices

  18. Karon86
    • one year ago
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    Dude. Like, That is totally different from what I am taught by the book

  19. amistre64
    • one year ago
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    i contend that it can be any value we want it to be if there are no other conditions for the sequence.

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  20. jim_thompson5910
    • one year ago
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    @Karon86 how does the book show you? what method do they use?

  21. Karon86
    • one year ago
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    To find the explicit formal, expand the first few terms of the sequence(sorry I have a terrible iPad): A1: 1 A2: 1+2 = 3

  22. Karon86
    • one year ago
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    A3: 1+2+3 = 6 A4: 1+2+3+4=10

  23. Karon86
    • one year ago
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    A5:1+2+3+4+5=15 An: 1+2+...+n

  24. Karon86
    • one year ago
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    Are you still here

  25. jim_thompson5910
    • one year ago
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    I see, let me think

  26. jim_thompson5910
    • one year ago
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    can you find an explicit formula for An: 1+2+...+n

  27. Karon86
    • one year ago
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    I don't know, the book doesn't says: Therefore: An=1+2+3+...+(n-2)+(n-1)+n. (this is whatI I am so confused) Which you can write as An=n+(n-1)+(n-2)+....+3+2+1

  28. Karon86
    • one year ago
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    Why is it that way?

  29. Karon86
    • one year ago
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    I am confused with what comes after therefore

  30. jim_thompson5910
    • one year ago
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    notice how the n pairs with 1, n-1 pairs with 2, n-3 with 3, etc etc

  31. jim_thompson5910
    • one year ago
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    if you were to add up those pairs you'd get n+1 = n+1 n-1+2 = n+1 n-2+3 = n+1 ... ... ... 3+n-2 = n+1 2+n-1 = n+1 1+n = n+1

  32. Karon86
    • one year ago
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    I don't understand

  33. jim_thompson5910
    • one year ago
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    this video might help make things click a bit better? https://www.youtube.com/watch?v=aaFrAFZATKU let me know if it doesn't

  34. Karon86
    • one year ago
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    I cannot acces the YouTube video because it won't let me on th iPad. Please forgive me. Could you tell me the name of the YouTube video

  35. jim_thompson5910
    • one year ago
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    do you have access to a laptop or desktop computer?

  36. Karon86
    • one year ago
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    No sir but if I can acces it on my phone. God I wish I had a laptop of desktop

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