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Math Help - A power question....

  1. #1
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    Smile A power question....

    12 means I x 1, 22 means 2 x 2, 32means 3 x 3, and so forth. 12+ 22+ 32+ 42 + ... + 252= 5525, and 22+ 42+ 62 + 82 + ... + 502 = N. Find the value of N.

    Now we have been told the answer is 22,100; but I don't know how to get there the fast way, apart from actually doing the equation fo each even number until I get to 50. And then adding them all up!

    Would anyone be able to explain to a me? Thank you.
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  2. #2
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    Quote Originally Posted by ronaldj View Post
    12 means I x 1, 22 means 2 x 2, 32means 3 x 3, and so forth. 12+ 22+ 32+ 42 + ... + 252= 5525, and 22+ 42+ 62 + 82 + ... + 502 = N. Find the value of N.

    Now we have been told the answer is 22,100; but I don't know how to get there the fast way, apart from actually doing the equation fo each even number until I get to 50. And then adding them all up!

    Would anyone be able to explain to a me? Thank you.

    Hint:- use
    \boxed { 1^2+2^2+3^2+4^2+...............+n^2= \frac{n(n+1)(2n+1)}{6} }
    see pattern in
    12+ 22+ 32+ 42 + ... + 252= 5525
    LHS =1*1+2*2+3*3+4*4+............+25*25
    =1^2+2^2+3^2+4^2+...............+25^2
     =\frac{25(25+1)(50+1)}{6}=5525 = RHS

    therefore
    22+ 42+ 62 + 82 + ... + 502 = N

     2^2+4^2+6^2+8^2+10^+.....+50^2
     =4+16+36+64+100+.......+2500
    =4(1+4+9+16+25+.........+625)
     =4(1^2+2^2+3^2+4^2+...............+25^2)
    = 4 \cdot \frac{25(25+1)(50+1)}{6} =22100
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  3. #3
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    Many thanks for taking the time to answer ! Appreciated.
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