Sequences and series (1)

Algebra Level 5

Find the sum of the terms of the Geometric Progression, \(a+ar+ar^2 + ar^3 + \cdots\), where \(a\) is the value of \(x\) for which the function \(7+2x \ln 25 -5^{x-1} - 5^{2-x}\) has a greatest value and \(r\) is equal to \( \displaystyle\lim_{x\rightarrow 0} \displaystyle\int_{0}^{x} \frac{t^2 dt}{x^2 \tan (π+x)}\)


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