please help wit the problem below

Question

prove that for every n∈N, ∑k²=(n(n+1)(2n+1))/6. Prove using mathematical induction

Answer 1

This is a standard problem provable by Mathematical Induction.  The M.I. method is easily summarized:

 

1.  Verify that this is true when n = 1.

2. ASSUME that this is true when n = k.

3. Using the formula with k substituted for n, add what would be the next term to each side of the equation.

4. Using your mastery of Algebra, put the left side under the control of a summation that runs from n = 1 to n = k + 1

5. When you find that the right side is equal the right side of the formula when k+1 is substituted there for n, you will have SHOWN THAT the formula applies for case k+1 whenever it applies for case k, AND SINCE we know that's true for k = 1, this implies (dictates) that it must be true for k =2, and furthermore this reasoning can be repeated (case 2 implies case 3 follows, etc. etc. ad infinitum).

 

Note: One's Algebra skills will have to be good, including factoring!