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Difficult sequence problem

  1. Apr 15, 2008 #1
    1. The problem statement, all variables and given/known data

    A sequence of terms U1, U2, U3, ... is defined by Un = 5n + 20

    If the sum of the first N terms of the sequence defined above equals the sum of the first kN natural numbers, show that:

    N = 45 - k / k^2 - 5

    2. Relevant equations

    Sum of first N natural numbers = N(N+1)/2

    Sum of sequence = N(50+(N-1)5)/2 (knowing that a=25, d=5)

    3. The attempt at a solution

    I put the substituted kN into N in the equation N(N+1)/2 to give me kN(kN+1)/2.

    I then made kN(kN+1)/2 = N(50+(N-1)5)/2 and solved from there which eventually got me to:

    k^2N^2 + kN = 50N + 5N^2 -5

    (I then divided both sides by N as N is always positive)

    N = 45 +5N -k / k^2

    ...the +5N makes it wrong. I dont know where I've gone wrong, any help would be appreciated.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Apr 15, 2008 #2
    Your last expression is right, the only thing you need to do is solve for N.
  4. Apr 15, 2008 #3
    This one should be k^2N^2 + kN = 50N + 5N^2 -5N, but I suppose this was an inconsequential typo.

    Dividing by N is OK, but this is not the result. Just divide the above with N (all the terms contain N so this is easy), and then group terms with and without N.
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