SOLUTION TO PROBLEM R-14

Since n, n+1, n² are the sides of a triangle, one has that necessarily

$$n^2 < n + (n+1) \Leftrightarrow n^2 - 2n < 1 \Leftrightarrow n^2 - 2n + 1 < 2$$

The last inequality can be written as

(n-1)² < 2

Since n-1 is a nonnegative integer, only n-1 = 0 and n - 1 = 1 would be acceptable choices, so n is 1 or 2.

n = 1 is not possible because n = 1, n+1 = 2, n² = 1 should then be the sides of a triangle for which reason one should have that 2 < 1 + 1 which is false.

On the other hand, n = 2 is OK because n+1 = 3 and n² = 4 and the following inequalities are correct 2 < 2+4, 3 < 2+4, 4 < 2+3. Hence the only possibility is n = 2.

Questions and/or comments should be directed to Judy Downey or Griff Elder

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Last modified:   Fri Apr 26 07:47:12 CDT 2002