   Formulate equations
  How do people formulate equations for sequences. For example :1 + 2 + ...n = n(n+1)/2We could prove this by mathmatical induction, but how does one arive at n(n+1)/2as a guess and what not?   
  For your given example, the solution is arrived at by some creative yet very simple alegbra. In this particular case, its done by writing the solution as 1+2+...+n=x, adding the equation to itself, and then solving for x after first recognizing that an equivalent way of writing 1+2+...+n is n+(n-1)+(n-2)+...+1.   ;  The average term of the sum 1 + 2 + ... + n is (n+1)/2. There are n terms. So the sum is n(n+1)/2.To evaluate the sum of squares 1^2 + 2^2 + ... + (n-1)^2 we consider instead the sum of cubes 1^3 + 2^3 + ... + (n-1)^3 + n^3. This is the sum of (k + 1)^3 for k = 0 to n - 1. Write (k + 1)^3 = k^3 + 3k^2 + 3k + 1, so we find n^3 = 3 * (the sum of k^2 for k = 1 to n - 1) + 3 * (n^2/2 - n/2) + n. Simplifying yields the sum of k^2 for k = 1 to n-1 is n^3/3 - n^2/2 + n/6.The procedure can be repeated to derive formulas for 1^j + 2^j + ... + n^j for all j. Alternatively, the general formula can be obtained from the Euler-Maclaurin summation formula, which brings in the Bernoulli numbers. Look here to see this done.[Edited by - nilkn on April 11, 2010 1:18:49 AM]