Question

Which term of the sequence $20,19\frac{1}{4},18\frac{1}{2},17\frac{3}{4},.....$ is the first negative term?

Answer 1

The given sequence is an A.P. in which first term $(a=20)$ and common difference $(d=-\frac{3}{4})$.

Let the $n^{th}$ term of the given A.P. be the first negative term.

Then, $a_n<0$

$⟹a+(n-1)d<0$

$⟹20+(n-1)\times (-\frac{3}{4})<0$

$⟹20-\frac{3(n-1)}{4}<0$

$⟹20-\frac{3n-3}{4}<0$

$⟹\frac{80-3n+3}{4}<0$

$⟹\frac{83-3n}{4}<0$

$⟹83-3n<0$

$⟹83<3n$

$⟹3n>83$

$⟹n>27\frac{2}{3}$

$⟹n\ge 28[\because n$ is a natural number]

Thus ${28}^{th}$ term of the given sequence is the first negative term.