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 $${S_1}^2 + {S_2}^2 + {S_3}^2 + ....... + {S^2}_{2n - 1}$$

Find the sum of the series : $$$\sum\limits_{r = 0}^n {{{\left( { - 1} \right)}^r}\,{}^n{C_r}\left[ {{1 \over {{2^r}}} + {{{3^r}} \over {{2^{2r}}}} + {{{7^r}} \over {{2^{3r}}}} + {{{{15}^r}} \over {{2^{4r}}}}..........up\,\,to\,\,m\,\,terms} \right]} $$$

If $$a > 0,\,b > 0$$ and $$\,c > 0,$$ prove that $$\,c > 0,$$ prove that $$\left( {a + b + c} \right)\left( {{1 \over a} + {1 \over b} + {1 \over c}} \right) \ge 9$$