The 20^th~ term from the end of the progression 20,1914,1812,1734,, - 12914 is :

20,19 14, 18 12, 17 34, .,-129 14 \\ This is A.P. with common difference \\ d_1=-1+14=-34 \\-129 14, 19 14, 20 \\\\ This is also A.P.

The 20^{{th}{~}} term from the end of the progression 20,19\frac{1}{4},18\frac{1}{2},17\frac{3}{4},, - 129\frac{1}{4} is :: Sequence & Series (Mathematics)

$20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots \ldots .,-129 \frac{1}{4} \\$

This is A.P. with common difference

$\\ \mathrm{d}_1=-1+\frac{1}{4}=-\frac{3}{4}$

$\\-129 \frac{1}{4}, \ldots \ldots \ldots \ldots \ldots 19 \frac{1}{4}, 20 \\\\$

This is also A.P. $\mathrm{a}=-129 \frac{1}{4}$ and $\mathrm{d}=\frac{3}{4}\\$

Required term = $-129 \frac{1}{4}+(20-1)\left(\frac{3}{4}\right)$ $\\=-129-\frac{1}{4}+15-\frac{3}{4} \\$

=-115