The question today comes from 1994 Paper I. The original question is shown below.
(i) Write down the average of the integers . Show that
(ii) Write down and prove a general law of which the following are special cases:
Of course, this question could be (reasonably) easily done by induction. But why always induction? Writing an induction argument is complicated and it does not reveal any information about the underlying hidden pattern. Why not try deduction?
Also, after doing part (ii) I would kindly ask the reader to find a general formula (or a recursive one) of where is a positive integer.
(i) The average is obviously . The sum is (also obviously) the average times the number of terms, which in this case is .
(ii) A general rule would be , or, written more compactly using the sigma notation,
Part (i) can be used in the proof of that summation. Noticing that the average is and there are terms, we can calculate the LHS as
For the last part, consider summing up all the equations, giving . The LHS is just while the RHS is , from which follows.