Answer :
To solve this problem, let's look at the sequence provided: 9, 19, 29, 39, 49, 59, ...
We need to determine which recursive formula can be used to define this sequence for [tex]\(n > 1\)[/tex].
1. Identify the Pattern:
- Start with the first term: 9
- Observe how each number relates to its predecessor:
- 19 is 10 more than 9,
- 29 is 10 more than 19,
- 39 is 10 more than 29,
- 49 is 10 more than 39,
- 59 is 10 more than 49.
This tells us that each term in the sequence is generated by taking the previous term and adding 10 to it.
2. Formulate the Recursive Rule:
- Let [tex]\( f(n) \)[/tex] be the nth term of the sequence.
- The relationship between the terms can be expressed as:
[tex]\[
f(n) = f(n-1) + 10
\][/tex]
This means each term [tex]\( f(n) \)[/tex] is equal to the previous term [tex]\( f(n-1) \)[/tex] plus 10.
3. Verify the Options:
- Now, compare this recursive formula with the given options:
- [tex]\( f(n) = 10 f(n-1) \)[/tex]
- [tex]\( f(n) = f(n-1) + f(n-2) + 10 \)[/tex]
- [tex]\( f(n) = f(n-1) + 10 \)[/tex]
- [tex]\( f(n) = \frac{19}{9} f(n-1) \)[/tex]
- The correct option is [tex]\( f(n) = f(n-1) + 10 \)[/tex].
Therefore, the recursive formula that defines the given sequence for [tex]\( n > 1 \)[/tex] is:
[tex]\[ f(n) = f(n-1) + 10 \][/tex]
We need to determine which recursive formula can be used to define this sequence for [tex]\(n > 1\)[/tex].
1. Identify the Pattern:
- Start with the first term: 9
- Observe how each number relates to its predecessor:
- 19 is 10 more than 9,
- 29 is 10 more than 19,
- 39 is 10 more than 29,
- 49 is 10 more than 39,
- 59 is 10 more than 49.
This tells us that each term in the sequence is generated by taking the previous term and adding 10 to it.
2. Formulate the Recursive Rule:
- Let [tex]\( f(n) \)[/tex] be the nth term of the sequence.
- The relationship between the terms can be expressed as:
[tex]\[
f(n) = f(n-1) + 10
\][/tex]
This means each term [tex]\( f(n) \)[/tex] is equal to the previous term [tex]\( f(n-1) \)[/tex] plus 10.
3. Verify the Options:
- Now, compare this recursive formula with the given options:
- [tex]\( f(n) = 10 f(n-1) \)[/tex]
- [tex]\( f(n) = f(n-1) + f(n-2) + 10 \)[/tex]
- [tex]\( f(n) = f(n-1) + 10 \)[/tex]
- [tex]\( f(n) = \frac{19}{9} f(n-1) \)[/tex]
- The correct option is [tex]\( f(n) = f(n-1) + 10 \)[/tex].
Therefore, the recursive formula that defines the given sequence for [tex]\( n > 1 \)[/tex] is:
[tex]\[ f(n) = f(n-1) + 10 \][/tex]
