# How do you tell whether the sequence 100, 50, 25, 25/2, 25/4,... is geometric?

Oct 31, 2017

The sequence is geometric with first term ${a}_{1} = 100$ and common ratio $r = 0.5$

#### Explanation:

${a}_{1} = 100 , {a}_{2} = 50 , {a}_{3} = 25 , {a}_{4} = \frac{25}{2} , {a}_{5} = \frac{25}{4.} . .$

$\frac{50}{100} = 0.5 , \frac{25}{50} = 0.5 , \frac{\frac{25}{2}}{25} = 0.5 , \frac{\frac{25}{4}}{\frac{25}{2}} = 0.5$

$\therefore {a}_{2} / {a}_{1} = {a}_{3} / {a}_{2} = {a}_{4} / {a}_{3} = {a}_{5} / {a}_{4} = 0.5$

The ratio of successive term and preceeding term is constant

$\left(r = 0.5\right)$ . Therefore the sequence is geometric whose first

term is ${a}_{1} = 100$ and common ratio is $r = 0.5$ [Ans]