Karon86
  • Karon86
What is the 100th term of sequence: 1,3,6,10,15,.... Write a explicit formula
Algebra
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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jim_thompson5910
  • jim_thompson5910
|dw:1443915861353:dw|
jim_thompson5910
  • jim_thompson5910
the jump from 1 to 3 is +2 |dw:1443915905024:dw|
jim_thompson5910
  • jim_thompson5910
the jump from 3 to 6 is +3 |dw:1443915932403:dw|

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jim_thompson5910
  • jim_thompson5910
the jump from 6 to 10 is +4 |dw:1443915954079:dw|
jim_thompson5910
  • jim_thompson5910
the jump from 10 to 15 is +5 |dw:1443915976631:dw|
jim_thompson5910
  • jim_thompson5910
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)
Karon86
  • Karon86
I want an explicit formula
jim_thompson5910
  • jim_thompson5910
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
jim_thompson5910
  • jim_thompson5910
|dw:1443916308731:dw|
jim_thompson5910
  • jim_thompson5910
the general quadratic is ax^2 + bx + c
jim_thompson5910
  • jim_thompson5910
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
jim_thompson5910
  • jim_thompson5910
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
jim_thompson5910
  • jim_thompson5910
|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
jim_thompson5910
  • jim_thompson5910
we now have this system of equations a+b+c = 1 4a+2b+c = 3 9a+3b+c = 6
Karon86
  • Karon86
I am still here
jim_thompson5910
  • jim_thompson5910
do you know how to solve for a,b,c?
jim_thompson5910
  • jim_thompson5910
you have a number of options 1) elimination 2) substitution 3) matrices
Karon86
  • Karon86
Dude. Like, That is totally different from what I am taught by the book
amistre64
  • amistre64
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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jim_thompson5910
  • jim_thompson5910
@Karon86 how does the book show you? what method do they use?
Karon86
  • Karon86
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
Karon86
  • Karon86
A3: 1+2+3 = 6 A4: 1+2+3+4=10
Karon86
  • Karon86
A5:1+2+3+4+5=15 An: 1+2+...+n
Karon86
  • Karon86
Are you still here
jim_thompson5910
  • jim_thompson5910
I see, let me think
jim_thompson5910
  • jim_thompson5910
can you find an explicit formula for An: 1+2+...+n
Karon86
  • Karon86
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
Karon86
  • Karon86
Why is it that way?
Karon86
  • Karon86
I am confused with what comes after therefore
jim_thompson5910
  • jim_thompson5910
notice how the n pairs with 1, n-1 pairs with 2, n-3 with 3, etc etc
jim_thompson5910
  • jim_thompson5910
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
Karon86
  • Karon86
I don't understand
jim_thompson5910
  • jim_thompson5910
this video might help make things click a bit better? https://www.youtube.com/watch?v=aaFrAFZATKU let me know if it doesn't
Karon86
  • Karon86
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
jim_thompson5910
  • jim_thompson5910
do you have access to a laptop or desktop computer?
Karon86
  • Karon86
No sir but if I can acces it on my phone. God I wish I had a laptop of desktop

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