If for an arithmetic progression $S_{2n} = 2S_n$,then $S_{3n} / S_n = \dots$

Let $\sum_{k=1}^{n} a_{k} = \alpha n^{2} + \beta n$. If $a_{10} = 59$ and $a_{6} = 7a_{1}$,then $\alpha + \beta$ is equal to:

If the sum of $n$ terms of an arithmetic progression is $3n^2 + 5n$ and $t_n = 164$,then $n = \dots$

In an $A.P.$,if the $p^{\text{th}}$ term is $\frac{1}{q}$ and the $q^{\text{th}}$ term is $\frac{1}{p}$,prove that the sum of the first $pq$ terms is $\frac{1}{2}(pq+1)$,where $p \neq q$.

If the numbers $a, b, c, d, e$ form an $A.P.$,then the value of $a - 4b + 6c - 4d + e$ is

$A$ farmer buys a used tractor for $Rs. 12000$. He pays $Rs. 6000$ cash and agrees to pay the balance in annual instalments of $Rs. 500$ plus $12\%$ interest on the unpaid amount. How much will the tractor cost him?