If ${a^{1/x}} = {b^{1/y}} = {c^{1/z}}$ and $a,\;b,\;c$ are in $G.P.$, then $x,\;y,\;z$ will be in

For any three positive real numbers $a,b,c$ ; $9\left( {25{a^2} + {b^2}} \right) + 25\left( {{c^2} - 3ac} \right) = 15b\left( {3a + c} \right)$ then

Let the sum of the first three terms of an $A. P,$ be $39$ and the sum of its last four terms be $178.$ If the first term of this $A.P.$ is $10,$ then the median of the $A.P.$ is

Let $V_{\mathrm{r}}$ denote the sum of the first $\mathrm{r}$ terms of an arithmetic progression $(A.P.)$ whose first term is $\mathrm{r}$ and the common difference is $(2 \mathrm{r}-1)$. Let

$T_{\mathrm{I}}=V_{\mathrm{r}+1}-V_{\mathrm{I}}-2 \text { and } \mathrm{Q}_{\mathrm{I}}=T_{\mathrm{r}+1}-\mathrm{T}_{\mathrm{r}} \text { for } \mathrm{r}=1,2, \ldots$

$1.$  The sum $V_1+V_2+\ldots+V_n$ is

$(A)$ $\frac{1}{12} n(n+1)\left(3 n^2-n+1\right)$

$(B)$ $\frac{1}{12} n(n+1)\left(3 n^2+n+2\right)$

$(C)$ $\frac{1}{2} n\left(2 n^2-n+1\right)$

$(D)$ $\frac{1}{3}\left(2 n^3-2 n+3\right)$

$2.$  $\mathrm{T}_{\mathrm{T}}$ is always

$(A)$ an odd number $(B)$ an even number

$(C)$ a prime number $(D)$ a composite number

$3.$  Which one of the following is a correct statement?

$(A)$ $Q_1, Q_2, Q_3, \ldots$ are in $A.P.$ with common difference $5$

$(B)$ $\mathrm{Q}_1, \mathrm{Q}_2, \mathrm{Q}_3, \ldots$ are in $A.P.$ with common difference $6$

$(C)$ $\mathrm{Q}_1, \mathrm{Q}_2, \mathrm{Q}_3, \ldots$ are in $A.P.$ with common difference $11$

$(D)$ $Q_1=Q_2=Q_3=\ldots$

Give the answer question $1,2$ and $3.$

Suppose we have an arithmetic progression $a_1, a_2, \ldots a_n, \ldots$ with $a_1=1, a_2-a_1=5$. The median of the finite sequence $a_1, a_2, \ldots, a_k$, where $a_k \leq 2021$ and $a_{k+1} > 2021$ is

If $^n{C_4},{\,^n}{C_5},$ and ${\,^n}{C_6},$ are in $A.P.,$ then $n$ can be