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Which term of the A.P. 20,77/4,37/2,75/4,..........is the first negative term?Find the term also. |
| Answer» The given AP is 20,\xa0{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 byan\xa0=a+(n-1)d<0{tex}\\Rightarrow{/tex}\xa0a+(n-1)d<0{tex}\\Rightarrow{/tex}\xa020+(n-1)({tex}\\frac{{ - 3}}{4}{/tex})<0{tex}\\Rightarrow{/tex}\xa020-{tex}\\frac{{3n}}{4}{/tex}+{tex}\\frac{3}{4}{/tex}<0{tex}\\Rightarrow{/tex}\xa0-\u200b\u200b\u200b\u200b\u200b\u200b{tex}\\frac{{3n}}{4}{/tex}+{tex}\\frac{{83}}{4}{/tex}<0{tex}\\Rightarrow{/tex}\xa0-3n+83<0{tex}\\Rightarrow{/tex}\xa0-3n< -83{tex}\\Rightarrow{/tex}\xa0n>{tex}\\frac{{83}}{3}{/tex}{tex}\\Rightarrow{/tex}\xa0n> 27.6So, n=28a28\xa0= a + 27d = 20 + 27\xa0{tex}\\times \\frac{-3}{4}{/tex}= -0.25Hence, the first negative term would be the 28th term.\xa0 | |