91Ó°ÊÓ

In a geometric sequence, each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general formula for any term in a geometric sequence is given by:
\(a_{n} = a \times r^{(n-1)}\),
where:

• \(a_{n}\) is the \(n\)-th term,
• \(a\) is the first term,
• \(r\) is the common ratio, and
• \(n\) is the term number.

This formula helps us find any term without listing all previous terms.
For our specific problem, the sequence is given by \(a_{n} = (-2)^n\). Notice that this formula already includes the common ratio \((-2)\) and assumes a first term equivalent to this value when \(n = 1\). It simplifies the process as we don't need to identify \(a\) and \(r\) explicitly.

To find a specific term in a geometric sequence, use the general formula and replace \(n\) with the term number you're interested in. In this case, to find the 5th term of our sequence, we start with the given general formula:
\(a_{n} = (-2)^n\).
For \(n = 5\), substitute 5 for \(n\):
\(a_{5} = (-2)^5\).
Next, compute the value:
Raising \((-2)\) to the 5th power involves multiplying \(-2\) by itself five times. Remember, multiplying a negative number by itself an odd number of times results in a negative number: \((-2)^5 = -32\).
So, the 5th term, \(a_5\), is \(-32\). Using this method, you can find any other term in the sequence by substituting the corresponding term number.

Exponents play a crucial role in geometric sequences, especially when dealing with negative bases and exponents. Negative exponents indicate the reciprocal of the base raised to the positive exponent, for instance:
\(a^{-n} = \frac{1}{a^n}\).
However, in this exercise, we're using positive exponents (but with a negative base \(a = -2\)).
To understand how negative bases affect the sequence:

• If the exponent is even, the result will be positive (e.g., \((-2)^2 = 4\)).
• If the exponent is odd, the result will be negative (e.g., \((-2)^3 = -8\)).

This alternating pattern continues with each step of the geometric sequence. In our problem, since all exponents are positive and increasing, alternate terms will alternate in sign: positive for even exponents and negative for odd ones. This helps predict whether a term in the sequence will be positive or negative based on the exponent.