Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
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.
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.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
