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Solution: Compute $ a_1 $ and $ a_4 $:
Question: An environmental scientist studying species richness in the Amazon finds that the number of species observed over time follows an arithmetic sequence. If the third years count is 25 and the seventh years count is 41, what is the total number of species observed over the first 10 years?
Solution: Let the first term be $ a $ and the common difference be $ d $.
Therefore, the minimum value of $ f(x) $ is:
This is a transcendental equation and difficult to solve analytically, so we evaluate $ g(t) $ numerically at key points:
Clearly, the minimum occurs at $ t = 0 $, giving minimum distance squared $ D(0) = 1 $, so the minimum distance is:
Solution: To find the value of $ m $ such that $ f(x) $ and $ g(x) $ intersect at exactly one point, set $ f(x) = g(x) $:
For the graphs to intersect at exactly one point, this quadratic must have exactly one real root. A quadratic has one real root when its discriminant $ D = 0 $. Rewrite the equation as: