Find the next three terms in each geometric sequence.

$1000,500,250\mathrm{.....}$

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, thesequence2, 6, 18, 54, ... is ageometric progressionwith common ratio 3.

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

The sequence:$1000,500,250\mathrm{.....}$

Each term in a geometric sequence can be expressed in terms of the first term $a_1$and the common ratio r. Since each succeeding term is formulated from one or more previous terms, this is a recursive formula.

First term $a_1=1000$and the common difference is:

$$\begin{gathered} r=\frac{a_2}{a_1} \\ =\frac{500}{1000} \\ =\frac{1}{2} \end{gathered}$$.

In order to calculate the next three terms of the geometric sequence substitute$n=4,5,6$ and 1000 for$a_1$ into the formula $a_n=a_1{r}^{n-1}$.

Thus, next three terms of the sequence$1000,500,250\mathrm{.....}$are$125,62.5,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}31.25}$.