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Answer :
To solve this problem, we want to determine which function among the given options generates the sequence: 26, 35, 44, 53, 62.
### Step-by-step Process:
1. Identify the Sequence Pattern:
Look at the differences between consecutive numbers:
- 35 - 26 = 9
- 44 - 35 = 9
- 53 - 44 = 9
- 62 - 53 = 9
Since the differences are constant, the sequence is arithmetic with a common difference of 9.
2. Form the General Expression:
For an arithmetic sequence, the nth term [tex]\(a_n\)[/tex] can be written as:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
Where:
- [tex]\(a_1\)[/tex] is the first term of the sequence,
- [tex]\(d\)[/tex] is the common difference,
- [tex]\(n\)[/tex] is the position in the sequence.
Here, [tex]\(a_1 = 26\)[/tex] and [tex]\(d = 9\)[/tex]. Plug these values in:
[tex]\[
a_n = 26 + (n-1) \cdot 9
\][/tex]
Simplify:
[tex]\[
a_n = 26 + 9n - 9 = 9n + 17
\][/tex]
3. Check the Given Options Against the Derived Formula:
We found the expression [tex]\(a_n = 9n + 17\)[/tex]. Review the options:
- A. [tex]\(f(n) = 9n + 17\)[/tex]
- B. [tex]\(\pi(n) = 17 - 9n\)[/tex]
- C. [tex]\(f(n) = 26.9n\)[/tex]
- D. [tex]\((f(n) = 0n + 17\)[/tex]
The correct function is option A, [tex]\(f(n) = 9n + 17\)[/tex].
This solution is consistent with the given sequence and clearly shows that selecting option A is the right answer.
### Step-by-step Process:
1. Identify the Sequence Pattern:
Look at the differences between consecutive numbers:
- 35 - 26 = 9
- 44 - 35 = 9
- 53 - 44 = 9
- 62 - 53 = 9
Since the differences are constant, the sequence is arithmetic with a common difference of 9.
2. Form the General Expression:
For an arithmetic sequence, the nth term [tex]\(a_n\)[/tex] can be written as:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
Where:
- [tex]\(a_1\)[/tex] is the first term of the sequence,
- [tex]\(d\)[/tex] is the common difference,
- [tex]\(n\)[/tex] is the position in the sequence.
Here, [tex]\(a_1 = 26\)[/tex] and [tex]\(d = 9\)[/tex]. Plug these values in:
[tex]\[
a_n = 26 + (n-1) \cdot 9
\][/tex]
Simplify:
[tex]\[
a_n = 26 + 9n - 9 = 9n + 17
\][/tex]
3. Check the Given Options Against the Derived Formula:
We found the expression [tex]\(a_n = 9n + 17\)[/tex]. Review the options:
- A. [tex]\(f(n) = 9n + 17\)[/tex]
- B. [tex]\(\pi(n) = 17 - 9n\)[/tex]
- C. [tex]\(f(n) = 26.9n\)[/tex]
- D. [tex]\((f(n) = 0n + 17\)[/tex]
The correct function is option A, [tex]\(f(n) = 9n + 17\)[/tex].
This solution is consistent with the given sequence and clearly shows that selecting option A is the right answer.
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