Which term of the progression 19,181/5,

 According to condition the given arithmetic progression is 19,18{tex}\frac{1}{5}{/tex},17{tex}\frac{2}{5}{/tex}.........(i)
Here, T₂ - T₁ = {tex}\frac { 91 } { 5 } - 19 = \frac { 91 - 95 } { 5 } = - \frac { 4 } { 5 }{/tex}
T₃ - T₂ = {tex}\frac { 87 } { 5 } - \frac { 91 } { 5 } = - \frac { 4 } { 5 }{/tex}
Therefore, (i) is an arithmetic progression with a = 19, d = {tex}-\frac{4}{5}{/tex}
Suppose, the nth term of the given arithmetic progression be the first negative term. Then, nth term < 0.
{tex}\Rightarrow{/tex} T_n < 0 

{tex}\Rightarrow{/tex} [ 19 + (n - 1){tex}\left( - \frac { 4 } { 5 } \right){/tex}] < 0
{tex}\Rightarrow{/tex} (99 - 4n) < 0

 {tex}\Rightarrow{/tex} 4n > 99

 {tex}\Rightarrow{/tex} n > {tex}24 \frac { 3 } { 4 }{/tex}.
{tex}\therefore{/tex} n = 25,

i.e., 25th is the first negative term in the given AP.