Question

In a geometric sequence, the term an+1 can be smaller than the term an.

true or false?

Answer 1

Do you mean a_(n+1), worded a sub (n+1)?

If so yes.  If the function of the sequence is getting smaller or more negative with each term.

Answer 2

Answer:

The given statement is true.

Step-by-step explanation:

In a geometric sequence the ratio of two consecutive terms should be equal. And the ratio of two consecutive terms is called the common ration.

For example 2,4,8,16.......represents geometric sequence because the ratio of consecutive terms are equal.

$\frac{4}{2}=\frac{8}{4}=\frac{16}{8}=2$

Here 2 is the common ratio.

There is no such conditions that the next term should be greater or smaller than the previous term. The condition is that ratio of consecutive terms should be equal.

Let us take another example.

For example 16,8,4,2.......represents geometric sequence because the ratio of consecutive terms are equal.

$\frac{8}{16}=\frac{4}{8}=\frac{2}{4}=\frac{1}{2}$

Here 1/2 is the common ratio.

In this example we can see that 2nd term is smaller than first term. 3rd term is smaller than second term.

Therefore, in general we can conclude that the term $a_{n+1}$can be smaller than the term $a_n$ in a geometric sequence

Hence, the given statement is true.