parmenides51 wrote: Let $\{a_n\}_{n \in N}$ be a sequence of real numbers with $a_1 = 2$ and $a_n =\frac{n^2 + 1}{\sqrt{n^3 - 2n^2 + n}}$ for all positive integers $n \ge 2$. Let $s_n = a_1 + a_2 + ...+ a_n$ for all positive integers $n$. Prove that $$\frac{1}{s_1s_2}+\frac{1}{s_2s_3}+ ...+\frac{1}{s_ns_{n+1}}<\frac15$$for all positive integers $n$. Easy to establish $a_n^2>n+2$, so $S_n>\sum_{k=1}^n\sqrt{n+2}>\frac 23n^{\frac 32}$ So $S_nS_{n+1}>S_n^2>\frac {4n^3}9$ And since $\sum_{k=n}^{+\infty}\frac 1{k^3}<\int_{n-1}^{+\infty}\frac{dx}{x^3}=\frac 1{2(n-1)^2}$ We get $\sum_{k=1}^{+\infty}\frac 1{S_kS_{k+1}}<\sum_{k=1}^n\frac 1{S_kS_{k+1}}+\frac 9{8n^2}$ Choosing in above inequality $n=4$ and computing explicitely $S_1,S_2,S_3,S_4,S_5$, we get $RHS<\frac 1{5}$, as required (note that choosing some less-rough approximations in previous steps could allow to compute explicitely less summands )