Final answer:
The rule for the sequence of the number of triangles as given is Tn = 3n - 2. This is calculated using the formula for an arithmetic sequence with a common difference of 3.
Step-by-step explanation:
The student is presented with a sequence of the number of triangles and is asked to write the rule for the sequence in terms of 'n'. From the given sequence of 1, 4, 7, and 10 triangles for the first four terms, it can be inferred that the number of triangles increases by 3 for each subsequent term. This pattern is an arithmetic sequence with a common difference of 3. To find the nth term of this sequence, the formula for an arithmetic sequence is used:
Tn = a + (n - 1)d
Where Tn is the nth term, a is the first term, n is the term number, and d is the common difference.
Since the first term a is 1 and the common difference d is 3, we substitute these values into the formula:
Tn = 1 + (n - 1)×3
Expanding this gives:
Tn = 1 + 3n - 3
Simplifying further, we get the rule for the nth term:
Tn = 3n - 2
This rule defines the number of triangles in the nth term of the given sequence. Consequently, for any given n, one can calculate the exact number of triangles present in that term of the sequence.