If the first term of an $A.P.$ is $3$ and the sum of its first four terms is equal to one-fifth of the sum of the next four terms,then the sum of the first $20$ terms is equal to

If $a, b, c, d, e$ are in $A.P.$,then the value of $a + b + 4c - 4d + e$ in terms of $a$,if possible,is:

The minimum value of ${\left( {\frac{3}{a} - 1} \right)^2} + {\left( {\frac{a}{b} - 1} \right)^2} + {\left( {\frac{b}{c} - 1} \right)^2} + {\left( {3c - 1} \right)^2}$ where $0 < a, b, c \leqslant 9$,is $p - q\sqrt{r}$; $p, q, r \in I$ and $q, r$ are coprime,then $(p + q + r)$ is equal to

Insert $6$ numbers between $3$ and $24$ such that the resulting sequence is an $A.P.$

If $\log 2, \log (2^n - 1)$ and $\log (2^n + 3)$ are in $A.P.$,then $n =$

The sum of the first four terms of an $A.P.$ is $56$. The sum of the last four terms is $112$. If its first term is $11$,then find the number of terms.