Consider this sequence:4, 11, 18, 25, . . .What is the explicit function that defines this sequence? If arithmetic, write the function in simplified form. If geometric, write the function with an exponent of n − 1.
To determine the explicit function that defines the given sequence, let’s analyze the differences between consecutive terms
To determine the explicit function that defines the given sequence, let’s analyze the differences between consecutive terms.
The difference between the first and second terms is 11 – 4 = 7.
The difference between the second and third terms is 18 – 11 = 7.
The difference between the third and fourth terms is 25 – 18 = 7.
We can observe that the differences between consecutive terms are constant, which suggests that this sequence is an arithmetic sequence.
To find the explicit function, we need to find the common difference (d) of the arithmetic sequence. From the differences we calculated earlier, we see that the common difference is 7.
Now, we can express the nth term of an arithmetic sequence with the formula:
an = a1 + (n – 1)d
where
an is the nth term,
a1 is the first term, and
d is the common difference.
In our given sequence, the first term (a1) is 4 and the common difference (d) is 7. Substituting these values into the formula, we have:
an = 4 + (n – 1) * 7
Simplifying this expression, we get the explicit function for the given sequence as:
an = 7n – 3
Therefore, the explicit function that defines this sequence is 7n – 3.
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