Let $(u_n)$ be a sequence $u_i\neq0$ and $$\frac{1}{u_1u_2}+\frac{1}{u_2u_3}+\frac{1}{u_3u_4}+\cdots+\frac{1}{u_{n-1}u_n}=\frac{n-1}{u_1u_n}$$ for all $n\geq3$ Prove that the sequence $(u_n)$ is arithmetic sequence
Prove that $u_n$ is arithmetic sequence if $\frac{1}{u_1u_2}+\frac{1}{u_2u_3}+\frac{1}{u_3u_4}+\cdots+\frac{1}{u_{n-1}u_n}=\frac{n-1}{u_1u_n}$
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sequences-and-series
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1I would start by making a spreadsheet of 10 to 20 terms, try some $u_1$'s and $u_2$'s and see if you get a hint from that. – 2012-04-03