Answer :
To determine the correct recursive formula for the sequence [tex]\(3, 24, 45, 66, 87, \ldots\)[/tex], let's break down the process step by step:
1. Identify the First Term:
- The first term of the sequence is [tex]\(3\)[/tex]. This will be important when writing the recursive formula.
2. Determine the Common Difference:
- Look at the differences between consecutive terms in the sequence:
- [tex]\(24 - 3 = 21\)[/tex]
- [tex]\(45 - 24 = 21\)[/tex]
- [tex]\(66 - 45 = 21\)[/tex]
- [tex]\(87 - 66 = 21\)[/tex]
- We observe that each term increases by [tex]\(21\)[/tex], so the common difference is [tex]\(21\)[/tex].
3. Formulate the Recursive Formula:
- A recursive formula expresses each term based on the previous one. With the first term as [tex]\(3\)[/tex] and a common difference of [tex]\(21\)[/tex], the formula becomes:
- [tex]\(f(1) = 3\)[/tex]
- [tex]\(f(n) = f(n - 1) + 21\)[/tex]
4. Select the Correct Option:
- Given the choices, the correct recursive formula that matches our findings is:
- A: [tex]\(f(1) = 3; f(n) = f(n - 1) + 21\)[/tex]
This option accurately describes the sequence, with each term derived from the previous term by adding [tex]\(21\)[/tex] and the first term being [tex]\(3\)[/tex].
1. Identify the First Term:
- The first term of the sequence is [tex]\(3\)[/tex]. This will be important when writing the recursive formula.
2. Determine the Common Difference:
- Look at the differences between consecutive terms in the sequence:
- [tex]\(24 - 3 = 21\)[/tex]
- [tex]\(45 - 24 = 21\)[/tex]
- [tex]\(66 - 45 = 21\)[/tex]
- [tex]\(87 - 66 = 21\)[/tex]
- We observe that each term increases by [tex]\(21\)[/tex], so the common difference is [tex]\(21\)[/tex].
3. Formulate the Recursive Formula:
- A recursive formula expresses each term based on the previous one. With the first term as [tex]\(3\)[/tex] and a common difference of [tex]\(21\)[/tex], the formula becomes:
- [tex]\(f(1) = 3\)[/tex]
- [tex]\(f(n) = f(n - 1) + 21\)[/tex]
4. Select the Correct Option:
- Given the choices, the correct recursive formula that matches our findings is:
- A: [tex]\(f(1) = 3; f(n) = f(n - 1) + 21\)[/tex]
This option accurately describes the sequence, with each term derived from the previous term by adding [tex]\(21\)[/tex] and the first term being [tex]\(3\)[/tex].
