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Find the product of this term

  1. Mar 16, 2014 #1

    utkarshakash

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    Gold Member

    1. The problem statement, all variables and given/known data
    {a_n} and {b_n} are two sequences given by [itex]a_n = (x)^{1/2^n}+(y)^{1/2^n} [/itex] and
    [itex]b_n = (x)^{1/2^n}-(y)^{1/2^n}[/itex]. Then find [itex]a_1 a_2 a_3 ..... a_n [/itex] in terms of x,y and b_n


    2. Relevant equations

    3. The attempt at a solution

    I tried substituting n=1,2,3 and so on for a few terms of a_n but really couldn't see anything important. I tried finding x and y instead in terms of a_n and b_n and got this

    [itex]x =\left( \dfrac{a_n + b_n}{2} \right) ^{2^n} \\
    y = \left( \dfrac{a_n - b_n}{2} \right) ^{2^n}[/itex]
     
  2. jcsd
  3. Mar 16, 2014 #2
    Observe that
    $$b_n=(x)^{1/2^n}-(y)^{1/2^n}=((x)^{1/2^{n+1}})^2-((y)^{1/2^{n+1}})^2$$
    That should give you a hint. :)
     
  4. Mar 16, 2014 #3

    utkarshakash

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    I can only think of using the identity x^2 - y^2 = (x+y)(x-y). But this does not help here :(
     
  5. Mar 16, 2014 #4
    It does help. Do you see that you should get ##b_n=b_{n+1}a_{n+1}## from your identity?
     
  6. Mar 16, 2014 #5

    utkarshakash

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    Nice approach Thanks!
     
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