Using induction or otherwise, prove that for any non-negative integers $$m$$, $$n$$, $$r$$ and $$k$$ ,
$$\sum\limits_{m = 0}^k {\left( {n - m} \right)} {{\left( {r + m} \right)!} \over {m!}} = {{\left( {r + k + 1} \right)!} \over {k!}}\left[ {{n \over {r + 1}} - {k \over {r + 2}}} \right]$$

Eighteen guests have to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. Determine the number of ways in which the sitting arrangements can be made.

Let p be the first of the n arithmetic means between two numbers and q the first of n harmonic means between the same numbers. Show that q does not lie between p and $$\,{\left( {{{n + 1} \over {n - 1}}} \right)^2}\,p$$.

If $${S_1}$$, $${S_2}$$, $${S_3}$$,.............,$${S_n}$$ are the sums of infinite geometric series whose first terms are 1, 2, 3, ...................,n and whose common ratios are $${1 \over 2}$$, $${1 \over 3}$$, $${1 \over 4}$$,....................$$\,{1 \over {n + 1}}$$ respectively, then find the values of $${S_1}^2 + {S_2}^2 + {S_3}^2 + ....... + {S^2}_{2n - 1}$$