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Arithmetic Progression problem

  1. Apr 14, 2014 #1

    utkarshakash

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    Gold Member

    1. The problem statement, all variables and given/known data

    Let a1,a2,a3........,a4001 are in A.P. such that [itex] \dfrac{1}{a_1a_2}+\dfrac{1}{a_2a_3}+.............................\dfrac{1}{a_{4000}a_{4001}} = 10 [/itex] and a2+a4000=50. Then |a1-a4001|


    3. The attempt at a solution

    [itex]\dfrac{1}{a_2} \left( \dfrac{1}{a_1} + \dfrac{1}{a_3} \right) + \dfrac{1}{a_4} \left( \dfrac{1}{a_3} + \dfrac{1}{a_5} \right) ........... \\

    2 \left( \dfrac{1}{a_1a_3} + \dfrac{1}{a_3a_5} ....... \right) [/itex]

    Similarly the number of terms will keep on reducing but it's really difficult to see manually what will I end up with?
     
  2. jcsd
  3. Apr 14, 2014 #2
    Hi utkarshakash! :smile:

    Try the following, I am not sure if this is going to work:
    $$\frac{1}{a_n a_{n+1}}=\frac{1}{d}\left(\frac{a_{n+1}-a_n}{a_n a_{n+1}}\right)=\frac{1}{d}\left(\frac{1}{a_n}-\frac{1}{a_{n+1}}\right)$$
    where ##d## is the common difference of AP.
     
  4. Apr 14, 2014 #3

    utkarshakash

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    Gold Member

    It does help. Thanks a lot!
     
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