In an $A.P.,$ the first term is $2$ and the sum of the first five terms is one-fourth of the next five terms. Show that $20^{th}$ term is $-112$

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?

If $n$ is the smallest natural number such that $n+2 n+3 n+\ldots+99 n$ is a perfect square, then the number of digits of $n^2$ is

If $x=\sum \limits_{n=0}^{\infty} a^{n}, y=\sum\limits_{n=0}^{\infty} b^{n}, z=\sum\limits_{n=0}^{\infty} c^{n}$, where $a , b , c$ are in $A.P.$ and $|a| < 1,|b| < 1,|c| < 1$, $abc \neq 0$, then

The sum of the first and third term of an arithmetic progression is $12$ and the product of first and second term is $24$, then first term is

The sum of the first $20$ terms common between the series $3 +7 + 1 1 + 15+ ... ......$ and $1 +6+ 11 + 16+ ......$, is