question_answer

Which term of the progression \[20,19\frac{1}{4},18\frac{1}{2},17\frac{3}{4},\] ?? is the first negative term?

Answer:

Given, A.P is \[20,19\frac{1}{4},18\frac{1}{2},17\frac{3}{4},\]???.

\[=20,\frac{77}{4},\frac{37}{2},\frac{71}{4},\]??

Here, \[a=20,d=\frac{77}{4}-20=\frac{77-80}{4}=\frac{-3}{4}\]

Let \[{{a}_{n}}\] is first negative term

\[\Rightarrow \]               \[{{a}_{n}}+(n-1)d<0\]

\[\Rightarrow \]        \[20+(n-1)\left( -\frac{3}{4} \right)<0\]

\[\Rightarrow \]              \[20-\frac{3}{4}n+\frac{3}{4}<0\]

\[\Rightarrow \]                     \[20+\frac{3}{4}<\frac{3}{4}n\]

\[\Rightarrow \]                           \[\frac{83}{4}<\frac{3}{4}n\]

\[\Rightarrow \]               \[n>\frac{83}{4}\times \frac{4}{3}\]

\[\Rightarrow \]               \[n>\frac{83}{4}=27.66\]

28th term will be first negative term of given A.P.