91Ó°ÊÓ

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

For example, the sequence $2,6,18,54,...$is a geometric progression with common ratio 3.

The ration of $2^{nd}$term and first term is called the common ratio.

The ball loses $\frac{1}{2}$its height after each bounce; hence it forms a geometric sequence. To determine a term (height after each bounce) multiply the previous term by $\frac{1}{2}$.

Initial height is the first term = 20 feet.

$1^{st}$term: $a_1=20$and the common difference is $\frac{1}{2}$.

$2^{nd}$term: $a_2=10$;

$3^{rd}$term: $a_3=5$

4^th term: $a_4=5\text{ }×\frac{1}{2}\text{ }=2.5$;

$5^{th}$ term: $a_5=2.5×\frac{1}{2}\text{ }=1.25$

$6^{th}$term: $a_6=1.25×\text{ }\frac{1}{2}\text{ }=0.625$ and so on.

The GP series thus obtained is $20,10,5,2.5,1.25,\mathrm{0.625.....}$. Plot these points to obtained the graph of the situation: