To determine whether the term [tex]\( a_{n+1} \)[/tex] can be smaller than the term [tex]\( a_n \)[/tex] in a geometric sequence, let's first understand the properties of a geometric sequence.
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, denoted as [tex]\( r \)[/tex].
Given:
- The [tex]\( n \)[/tex]-th term of the sequence is denoted as [tex]\( a_n \)[/tex]
- The [tex]\((n + 1)\)[/tex]-th term of the sequence is denoted as [tex]\( a_{n+1} \)[/tex]
- The common ratio is [tex]\( r \)[/tex]
The relationship between consecutive terms in a geometric sequence can be expressed as:
[tex]\[ a_{n+1} = a_n \cdot r \][/tex]
Now, let's analyze the condition under which [tex]\( a_{n+1} \)[/tex] can be smaller than [tex]\( a_n \)[/tex]:
1. When the common ratio [tex]\( r \)[/tex] is less than 1 (but positive):
- If [tex]\( 0 < r < 1 \)[/tex], each multiplication with [tex]\( r \)[/tex] will result in a smaller number:
[tex]\[ a_{n+1} = a_n \cdot r \][/tex]
Since [tex]\( r \)[/tex] is a fraction less than 1, [tex]\( a_{n+1} \)[/tex] will be less than [tex]\( a_n \)[/tex].
2. When the common ratio [tex]\( r \)[/tex] is negative:
- If [tex]\( r \)[/tex] is a negative number (e.g., [tex]\( r < 0 \)[/tex]), the terms alternate their signs. If [tex]\( |r| < 1 \)[/tex], even in sign alternation, the absolute value diminishes:
[tex]\[ a_{n+1} = a_n \cdot r \][/tex]
For negative [tex]\( r \)[/tex] with [tex]\( |r| < 1 \)[/tex], if [tex]\( a_n \)[/tex] is positive, [tex]\( a_{n+1} \)[/tex] becomes negative and smaller in magnitude.
In conclusion, the condition when [tex]\( a_{n+1} \)[/tex] is smaller than [tex]\( a_n \)[/tex] is satisfied when the common ratio [tex]\( r \)[/tex] is either a fraction between 0 and 1 or a negative number with an absolute value less than 1. Hence, it is indeed true that:
[tex]\[ a_{n+1} \text{ can be smaller than } a_n \text{ in a geometric sequence.} \][/tex]
Therefore, the answer is:
True.
