If \(\vec{a}, \vec{b} \& \vec{c}\) are unit vectors such that \((\vec{a}-\vec{b})^2 + (\vec{b}-\vec{c})^2 + (\vec{c}-\vec{a})^2 = 9\). Find positive k if \(|2\vec{a} + k\vec{b} + k\vec{c}| = 3\) :
Let \(25^x+25^{-x},\ \dfrac{\alpha}{3},\ 20^{1+x}+20^{1-x}\), where \(x,\alpha\in\mathbb{R}\), be the first three terms of an A.P. of increasing terms. For the least value of \(\alpha\), the sum of its first \(10\) terms is _____