If the $p^{th}$ term of an arithmetic progression is $q$ and the $q^{th}$ term is $p$,then its $r^{th}$ term is:

The sum of all two-digit positive numbers which,when divided by $7$,yield $2$ or $5$ as a remainder is:

Let the positive integers be written in the form:
$1$
$2$ $3$
$4$ $5$ $6$
$7$ $8$ $9$ $10$
If the $k^{\text{th}}$ row contains exactly $k$ numbers for every natural number $k$,then the row in which the number $5310$ will be,is:

The sum of the first $p, q,$ and $r$ terms of an $A.P.$ are $a, b,$ and $c,$ respectively. Prove that $\frac{a}{p}(q-r)+\frac{b}{q}(r-p)+\frac{c}{r}(p-q)=0$.

If $1, \log _9(3^{1-x}+2), \log _3(4 \cdot 3^x-1)$ are in $A.P.$, then $x$ equals

The ratio of the sums of $m$ and $n$ terms of an $A.P.$ is $m^{2}: n^{2}$. Show that the ratio of the $m^{th}$ and $n^{th}$ term is $(2m-1):(2n-1)$.