A_2

Definition

In the context of geometric sequences, 'a_2' represents the second term of the sequence. The second term is calculated by multiplying the first term, 'a_1', by the common ratio, 'r', of the sequence.

5 Must Know Facts For Your Next Test

1. The formula for the second term of a geometric sequence is 'a_2 = a_1 \cdot r', where 'a_1' is the first term and 'r' is the common ratio.
2. The value of 'a_2' determines the behavior of the sequence, whether it is increasing, decreasing, or constant.
3. Knowing the values of 'a_1' and 'r' allows you to calculate any subsequent term in the sequence using the formula 'a_n = a_1 \cdot r^{n-1}'.
4. The ratio between consecutive terms in a geometric sequence is always constant, and this ratio is equal to the common ratio, 'r'.
5. The second term, 'a_2', is a crucial element in understanding the overall pattern and behavior of a geometric sequence.

Review Questions

• Explain the relationship between the first term ('a_1') and the second term ('a_2') in a geometric sequence.
  • In a geometric sequence, the second term, 'a_2', is calculated by multiplying the first term, 'a_1', by the common ratio, 'r'. This means that the value of 'a_2' is directly dependent on the value of 'a_1' and the common ratio. The relationship between 'a_1' and 'a_2' is crucial in determining the overall behavior of the sequence, as the second term sets the pattern for the subsequent terms.
• Describe how the value of 'a_2' affects the behavior of a geometric sequence.
  • The value of the second term, 'a_2', is a key factor in determining the behavior of a geometric sequence. If 'a_2 > a_1', the sequence is increasing, meaning each subsequent term will be larger than the previous one. If 'a_2 < a_1', the sequence is decreasing, with each term being smaller than the previous one. If 'a_2 = a_1', the sequence is constant, with all terms being equal. Understanding the relationship between 'a_1' and 'a_2' is crucial in predicting the overall pattern and behavior of a geometric sequence.
• Analyze the role of the common ratio, 'r', in the calculation of 'a_2' and the overall sequence.
  • The common ratio, 'r', is a fundamental component in the calculation of the second term, 'a_2', in a geometric sequence. The formula 'a_2 = a_1 \cdot r' demonstrates that the value of 'a_2' is directly proportional to the common ratio. The common ratio determines whether the sequence is increasing, decreasing, or constant, and it also allows for the calculation of any subsequent term in the sequence using the formula 'a_n = a_1 \cdot r^{n-1}'. Understanding the significance of the common ratio and its relationship to 'a_2' is essential in analyzing and working with geometric sequences.

Related terms

Geometric Sequence:

A sequence where each term is obtained by multiplying the previous term by a constant, known as the common ratio.

Common Ratio:

The constant ratio between consecutive terms in a geometric sequence.

First Term: The initial term in a sequence, denoted as 'a_1'.