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Thus, the complete factorization over the reals is:
Question: An ornithologist observes that the number of birds returning to a nesting site each year follows a sequence: $ a, a+d, a+2d, a+3d, a+4d $, forming an arithmetic sequence. If the sum of the squares of these five terms is 225 and the middle term is 5, find the value of $ a $.
Solution: The sequence is $ a, a+d, a+2d, a+3d, a+4d $. The middle term is $ a+2d = 5 $, so:
From (1): $ a = 5 - 2d $. Substitute into (2):
Question: An ichthyologist studying coral reef fish populations models the population trend with the equation $