91Ó°ÊÓ

The general term of a geometric sequence allows us to find any term in the sequence without having to list all the terms one-by-one. For a geometric sequence, the general term is expressed by the formula: \[a_n = a \times r^{(n-1)}\].
This formula essentially tells us how each term in the sequence is generated from the first term and the common ratio.
In this example, we found that the first term \(a\) is 2 and the common ratio \(r\) is also 2. By plugging these values into the general term formula, we get: \[a_n = 2 \times 2^{(n-1)}\].
This means that any term in the sequence can be found if we know its position \(n\). For example, to find the 5th term, just substitute \(n = 5\). That gives us:\[a_5 = 2 \times 2^{(5-1)} = 2 \times 16 = 32\].

So, the 5th term in this geometric sequence is 32.

The common ratio in a geometric sequence is what you multiply each term by to get to the next term. It's a constant value throughout the sequence.
In the given sequence, \(2, 4, 8, \text{...}\), we find the common ratio \(r\) by dividing the second term by the first term:\[\frac{4}{2} = 2\].
This tells us that each term is twice the term before it.
If we had a different sequence, like \(3, 6, 12, \text{...}\), the common ratio \(r\) would be: \[\frac{6}{3} = 2\], still 2, but with a different first term.
Identifying the common ratio is crucial because it is used in the general term formula to find any term in the sequence.

The first term of a geometric sequence is the starting point of the sequence. It's critical because the entire sequence is based on this initial value.
In the sequence \(2, 4, 8, \text{...}\), the first term \(a\) is 2.
This means our sequence starts at 2, and then each subsequent term is obtained by multiplying by the common ratio, which in this case is also 2.
If the first term changes, the sequence will look quite different even with the same common ratio.
For example, if the first term was 3 with the same common ratio of 2, the sequence would be \(3, 6, 12, 24, \text{...}\).
So, the first term sets the stage for the entire sequence and is always a necessary component in the general term formula \(a_n = a \times r^{(n-1)}\).