A_3

Definition

In the context of geometric sequences, 'a_3' represents the third term of the sequence. Geometric sequences are defined by a common ratio, 'r', which is used to generate each successive term from the previous one. The 'a_3' term is calculated by applying the common ratio to the first two terms of the sequence.

5 Must Know Facts For Your Next Test

1. The 'a_3' term in a geometric sequence is the third term, calculated by applying the common ratio to the first two terms.
2. The explicit formula for the 'n'th term of a geometric sequence is 'a_n = a_1 \cdot r^{n-1}', where 'a_1' is the first term and 'r' is the common ratio.
3. The 'a_3' term can be expressed as 'a_3 = a_1 \cdot r^2', where 'a_1' is the first term and 'r' is the common ratio.
4. The value of 'a_3' is crucial for understanding the behavior and patterns of a geometric sequence, as it helps determine the rate of growth or decay.
5. Identifying the 'a_3' term can assist in recognizing whether a sequence is geometric and in calculating other terms in the sequence.

Review Questions

• Explain how the 'a_3' term is related to the common ratio in a geometric sequence.
  • The 'a_3' term in a geometric sequence is directly related to the common ratio, 'r'. Specifically, the 'a_3' term is calculated by applying the common ratio twice to the first term, 'a_1'. The formula for the 'a_3' term is 'a_3 = a_1 \cdot r^2', where 'r' is the common ratio. This relationship demonstrates how the common ratio exponentially affects the growth or decay of the sequence, and the 'a_3' term provides insight into the overall behavior of the geometric sequence.
• Describe how the 'a_3' term can be used to identify the nature of a geometric sequence.
  • The value of the 'a_3' term can be used to determine whether a sequence is growing or decaying, as well as the rate of growth or decay. If 'a_3 > a_2', then the sequence is growing, and if 'a_3 < a_2', then the sequence is decaying. Additionally, the ratio between 'a_3' and 'a_2' is equal to the common ratio, 'r'. By analyzing the 'a_3' term in relation to the first two terms, you can identify the common ratio and understand the underlying pattern of the geometric sequence.
• Explain how the 'a_3' term can be used to calculate other terms in a geometric sequence.
  • The 'a_3' term can be used as a reference point to calculate any other term in a geometric sequence using the explicit formula, 'a_n = a_1 \cdot r^{n-1}'. Specifically, if you know the values of 'a_1' and 'a_3', you can solve for the common ratio, 'r', by dividing 'a_3' by 'a_1 \cdot r'. Once the common ratio is known, you can use the explicit formula to find the value of any other term, 'a_n', by plugging in the appropriate values for 'a_1', 'r', and 'n'. This demonstrates the importance of the 'a_3' term in understanding and working with geometric sequences.

Related terms

Geometric Sequence: A sequence where each term is a constant multiple of the previous term, known as the common ratio.

Common Ratio: The constant ratio between consecutive terms in a geometric sequence, denoted as 'r'.

Explicit Formula: The formula used to calculate any term in a geometric sequence, given the first term and the common ratio.