# If the common ratio of an infinite G.P be less than 1/2, show that each term will be greater than the sum of all the terms that follows it.

Let us represent the G.P. as :

$a,a{r}^{2},a{r}^{3},a{r}^{4},a{r}^{5},..................\phantom{\rule{0ex}{0ex}}Where,\phantom{\rule{0ex}{0ex}}a=Firstterm,r=commonratio\phantom{\rule{0ex}{0ex}}Accordingtothequestion,\phantom{\rule{0ex}{0ex}}r\frac{1}{2}\phantom{\rule{0ex}{0ex}}SumofinfiniteG.P.=\frac{a}{1-r}\phantom{\rule{0ex}{0ex}}Thedenominator(1-r)(-\frac{1}{2})\phantom{\rule{0ex}{0ex}}Now,letustakear\xb2asthefirstterm,then\phantom{\rule{0ex}{0ex}}Sum=\frac{ar\xb2}{1-r},\mathrm{sin}ce(1-r)(-\frac{1}{2})\phantom{\rule{0ex}{0ex}}Suma\phantom{\rule{0ex}{0ex}}Similarlyifwetakethefirsttermasa{r}^{3},then\phantom{\rule{0ex}{0ex}}sum\hspace{0.17em}ar\xb2\phantom{\rule{0ex}{0ex}}Andsoon$

Hence, we can say that each term is greater then sum of all the terms that follows it.

Regards

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