To determine who is correct, let's analyze the sequence and the formulas provided by Maxim and Salma.
The sequence given is: 54, 63, 72, 81, ...
First, we can observe the pattern in the sequence. Each term increases by 9:
- 63 - 54 = 9
- 72 - 63 = 9
- 81 - 72 = 9
This indicates that the sequence is an arithmetic sequence with a common difference of 9.
The first term \( f(1) \) is 54.
Now, the general formula for an arithmetic sequence can be expressed as:
\[ f(n) = a + (n - 1) \cdot d \]
where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.
Here, \( a = 54 \) and \( d = 9 \).
Substituting these values into the formula gives:
\[ f(n) = 54 + (n - 1) \cdot 9 \]
Simplifying this we get:
\[ f(n) = 54 + 9n - 9 = 9n + 45 \]
Let's check both formulas provided:
-
Maxim's formula: \( f(n) = 54 + 9n \)
- For \( n = 1 \): \( f(1) = 54 + 9 \cdot 1 = 54 + 9 = 63 \) (incorrect for first term)
- For \( n = 2 \): \( f(2) = 54 + 9 \cdot 2 = 54 + 18 = 72 \) (incorrect)
-
Salma's formula: \( f(n) = 9 + 54n \)
- For \( n = 1 \): \( f(1) = 9 + 54 \cdot 1 = 9 + 54 = 63 \) (incorrect for first term)
- For \( n = 2 \): \( f(2) = 9 + 54 \cdot 2 = 9 + 108 = 117 \) (incorrect)
Neither provided formula is correct as both yield wrong results for the first term or subsequent terms of the sequence.
The correct formula should be \( f(n) = 9n + 45 \). Thus, neither Maxim nor Salma is correct.