MathLabs

Problem 1

Determine all sequences of real numbers (a1,a2,…,a1995)(a_1,a_2,\ldots,a_{1995}) satisfying 2an−(n−1)≥an+1−(n−1)2\sqrt{a_n-(n-1)}\ge a_{n+1}-(n-1) for n=1,2,…,1994n=1,2,\ldots,1994, and 2a1995−1994≥a1+12\sqrt{a_{1995}-1994}\ge a_1+1.
Step 1 of 4: Shift the variables and record the domain
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xn=an−(n−1)≥0(1≤n≤1995)x_n=a_n-(n-1)\ge0\qquad(1\le n\le1995)
Detailed analysis

Set xn=an−(n−1)x_n=a_n-(n-1). The square roots in the statement imply xn≥0x_n\ge0 for every nn. The first inequalities become 2xn≥xn+1+12\sqrt{x_n}\ge x_{n+1}+1 for n<1995n<1995, while the last becomes 2x1995≥x1+12\sqrt{x_{1995}}\ge x_1+1.