If the $A.M., G.M.$ and $H.M.$ between two positive numbers $a$ and $b$ are equal, then

If $a,\;b,\;c$ are in $G.P.$, $a - b,\;c - a,\;b - c$ are in $H.P.$, then $a + 4b + c$ is equal to

Let $p =99$ and $q =101$. Define $p _1=\log \left(\frac{ p + q }{2}\right)$ and $q _1=\frac{1}{2}(\log p+\log q)$ and $p _2=\log \left(\frac{ p _1+ q _1}{2}\right), \quad q _2=\frac{1}{2}\left(\log p _1+\log q _1\right).$ Where all logarithms have base $10$ . Then

Let $n \geq 3$ be an integer. For a permutation $\sigma=\left(a_1, a_2, \ldots, a_n\right)$ of $(1,2, \ldots, n)$ we let $f_\sigma(x)=a_n x^{n-1}+a_{n-1} x^{n-2}+\ldots a_2 x+a_1$. Let $S_\sigma$ be the sum of the roots of $f_\sigma(x)=0$ and let $S$ denote the sum over all permutations $\sigma$ of $(1,2, \ldots, n)$ of the numbers $S_\sigma$. Then,

If the ratio of two numbers be $9:1$, then the ratio of geometric and harmonic means between them will be

If $A$ and $G$ be $A . M .$ and $G .M .,$ respectively between two positive numbers, prove that the numbers are $A \pm \sqrt{( A + G )( A - G )}$