91Ó°ÊÓ

In a geometric sequence, the common ratio is a crucial element that defines how the sequence progresses from one term to the next. The common ratio, denoted as \( r \), is the factor by which we multiply one term to obtain the next term. In simpler terms, if you know a term in the sequence, you multiply it by the common ratio to find the subsequent term.
For example, if the common ratio is 2 and you know the first term, let's say 3, then the second term is \( 3 \times 2 = 6 \). Likewise, the third term would be 6 times the common ratio, again 2, which equals 12.
Having a positive, constant common ratio means the sequence is increasing, while a negative common ratio can lead to alternating positive and negative terms.

The sequence formula in a geometric sequence is paramount for calculating any term within the sequence. The formula is given by \( a_n = a_1 \times r^{n-1} \), where:

• \( a_n \) is the nth term of the sequence.
• \( a_1 \) represents the first term.
• \( r \) is the common ratio.
• \( n \) is the term number.

This formula helps us predict any term if the first term and common ratio are known. For example, with \( a_1 = -3 \) and \( r = 2 \), the third term can be calculated without direct iteration: \( a_3 = -3 \times 2^{3-1} = -3 \times 4 = -12 \). This approach can make finding any term straightforward and efficient.

Calculating terms in a geometric sequence requires understanding both the sequence formula and the common ratio. To find any term, apply the sequence formula \( a_n = a_1 \times r^{n-1} \).

• For the first term, simply use \( a_1 \) itself.
• The second term \( a_2 = a_1 \times r \).
• For further terms, use the general formula for more complex sequences.

A practical example: if \( a_1 = -3 \) and \( r = 2 \), the first five terms calculated using the formula are:

• \( a_1 = -3 \)
• \( a_2 = -6 \)
• \( a_3 = -12 \)
• \( a_4 = -24 \)
• \( a_5 = -48 \)

Knowing how to calculate these can aid in quickly finding terms without having to go through multiple steps.

A geometric series is closely related to a geometric sequence. While a sequence lists numbers in order, a series sums them up. In a geometric series, you sum the terms of a geometric sequence.
If you have a finite sequence, such as the first five terms, the series would be the sum of those: \( S_n = a_1 + a_2 + a_3 + a_4 + a_5 \). This would be \(-3 - 6 - 12 - 24 - 48 = -93\).
For an infinite sequence, the series converges to a sum if the common ratio is between -1 and 1, thanks to the formula \( S = \frac{a_1}{1-r} \). Understanding how to sum geometric sequences can be valuable both in mathematics and real-world applications.