Question

define the sequence explicitly
7,11, 15,19

Answer 1

The given sequence is: 7, 11, 15, 19.

To define the sequence explicitly, we can observe that it is an arithmetic sequence where each term after the first is generated by adding a fixed number (the common difference) to the previous term.

In this case, the common difference is:

11 - 7 = 4,
15 - 11 = 4,
19 - 15 = 4.

The first term \( a_1 \) is 7, and the common difference \( d \) is 4.

The general formula for the \( n \)-th term \( a_n \) of an arithmetic sequence can be written as:

\[
a_n = a_1 + (n - 1) \cdot d
\]

Substituting \( a_1 = 7 \) and \( d = 4 \):

\[
a_n = 7 + (n - 1) \cdot 4
\]

Simplifying this expression gives:

\[
a_n = 7 + 4n - 4 = 4n + 3
\]

Thus, the explicit definition of the sequence is:

\[
a_n = 4n + 3
\]

for \( n = 1, 2, 3, \ldots \)