In a geometric sequence, the term a_n+1 can be smaller than the term a_n. True or False?

Solution:

Given, in a geometric sequence, the term a_n+1 can be smaller than the term a_n.

We have to determine if the given statement is true or false.

A geometric sequence is a special type of sequence where the ratio of every two successive terms is a constant. This ratio is known as a common ratio of the geometric sequence.

So a geometric sequence is in the form a, ar, ar²...

where 'a' is the first term

'r' is the common ratio of the sequence.

The common ratio can be either a positive or a negative real number.

Example: 3, 9, 27, 81,....

Here, a = 3

a_n+1 = 9

Common ratio, r = 9/3 = 27/9 = 81/27 = 3

Example: 81, 27, 9, 3,...

Here, a = 81

a_n+1 = 27

Common ratio, r = 81/27 = 27/9 = 9/3 = 3

We observe that the a_n+1 term is smaller than the term a_n.

Therefore, the term a_n+1 can be smaller than the term a_n.

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In a geometric sequence, the term a_n+1 can be smaller than the term an. True or False?

Summary:

The given statement,”In a geometric sequence, the term a_n+1 can be smaller than the term a_n” is true.