Which term of the A.P. 20,77/4,37/2,75/4,..........is …

The given AP is 20, {tex}19 \frac { 1 } { 4 } , 18 \frac { 1 } { 2 } , 17 \frac { 3 } { 4 } , \dots{/tex}
common difference ={tex}19 \frac { 1 } { 4 } - 20 = \frac { 77 } { 4 } - 20 = \frac { - 3 } { 4 }{/tex}
the general term of an AP is given by
a_n =a+(n-1)d<0
{tex}\Rightarrow{/tex} a+(n-1)d<0
{tex}\Rightarrow{/tex} 20+(n-1)({tex}\frac{{ - 3}}{4}{/tex})<0
{tex}\Rightarrow{/tex} 20-{tex}\frac{{3n}}{4}{/tex}+{tex}\frac{3}{4}{/tex}<0
{tex}\Rightarrow{/tex} -​​​​​​{tex}\frac{{3n}}{4}{/tex}+{tex}\frac{{83}}{4}{/tex}<0
{tex}\Rightarrow{/tex} -3n+83<0
{tex}\Rightarrow{/tex} -3n< -83
{tex}\Rightarrow{/tex} n>{tex}\frac{{83}}{3}{/tex}

{tex}\Rightarrow{/tex} n> 27.6
So, n=28
a₂₈ = a + 27d = 20 + 27 {tex}\times \frac{-3}{4}{/tex}= -0.25
Hence, the first negative term would be the 28th term.