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

If $a,\;b,\;c$ are in $A.P.$ and $a,\;b,\;d$ in $G.P.$, then $a,\;a - b,\;d - c$ will be in

If $a + 2b + 3c = 6$, then the greatest value of $abc^2$ is (where $a,b,c$ are positive real numbers)

Three numbers form a $G.P.$ If the ${3^{rd}}$ term is decreased by $64$, then the three numbers thus obtained will constitute an $A.P.$ If the second term of this $A.P.$ is decreased by $8$, a $G.P.$ will be formed again, then the numbers will be

Given a sequence of $4$ numbers, first three of which are in $G.P.$ and the last three are in $A.P$. with common difference six. If first and last terms in this sequence are equal, then the last term is

If $a,\;b,\;c$ be in $A.P.$ and $b,\;c,\;d$ be in $H.P.$, then