91Ó°ÊÓ

An arithmetic sequence is defined by the formula \( a_n = a_1 + (n - 1)d \), where \(a_n\) represents the \(n\)-th term, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term position. A geometric sequence is defined by the formula \( a_n = a_1 \times r^{n - 1} \), where \(a_n\) represents the \(n\)-th term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term position.

Since we are looking for sequences that are both arithmetic and geometric, the common difference of an arithmetic sequence and the common ratio of a geometric sequence should have a relationship. Let's set up these equations for consecutive terms. From the arithmetic sequence definition: \(a_2 - a_1 = d \) and \(a_3 - a_2 = d\). From the geometric sequence definition: \(a_2 = a_1 \times r\) and \(a_3 = a_2 \times r\).

Now, we want to find the relationship between \(d\) and \(r\). We can do this by expressing the common difference in terms of the common ratio using the equations we set up in step 2. Given that the sequence is both arithmetic and geometric, we know that: \(a_2 - a_1 = a_3 - a_2 \) So, \(a_1 \times r - a_1= a_2 \times r - a_2\) Divide by \(a_1\) to get: \(r - 1= (a_2/a_1) \times r - 1\).

Now we need to solve for the common ratio, \(r\), that will make the sequence both arithmetic and geometric. From the equation in step 3, we find that: \(r - 1= r \times (a_2/a_1) - 1\) To find \(r\), we can rewrite the equation as: \(r - 1 = (r - 1) \times (a_2/a_1)\). If \(r - 1\) is not equal to 0 (meaning \(r \neq 1\)), we can divide by it: \(1 = a_2/a_1\) This means that \( a_2 = a_1\). In other words, the sequence is constant and it satisfies both conditions (arithmetic and geometric). On the other hand, if \(r - 1 = 0\) (meaning \(r = 1\)), the sequence can also be both arithmetic and geometric, as any term multiplied by \(r=1\) will yield the same term, and the common difference will always be zero.

We found that all infinite sequences that are both arithmetic and geometric are constant sequences. Constant sequences satisfy the arithmetic condition because the difference between consecutive terms is always zero, and they satisfy the geometric condition because the common ratio is always 1.