In the context of geometric sequences, $a_4$ refers to the fourth term of the sequence. Geometric sequences are a type of sequence where each term is a constant multiple of the previous term, creating a pattern of growth or decay over time.
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$a_4$ is the fourth term in a geometric sequence, following $a_1$, $a_2$, and $a_3$.
The value of $a_4$ can be calculated using the explicit formula for a geometric sequence: $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.
The common ratio, $r$, is the constant multiplier between consecutive terms in the geometric sequence.
Knowing the values of $a_1$ and $r$ allows you to determine the value of $a_4$ and any other term in the sequence.
Understanding the relationship between $a_4$ and the other terms in the sequence is crucial for solving problems involving geometric sequences.
Review Questions
Explain how the value of $a_4$ is related to the first term, $a_1$, and the common ratio, $r$, in a geometric sequence.
The value of $a_4$ in a geometric sequence is determined by the first term, $a_1$, and the common ratio, $r$, using the explicit formula $a_n = a_1 \cdot r^{n-1}$. Specifically, $a_4 = a_1 \cdot r^{3}$, where $r$ is the constant multiplier between consecutive terms in the sequence. By knowing the values of $a_1$ and $r$, you can calculate the value of $a_4$ and any other term in the geometric sequence.
Describe how the position of $a_4$ within the sequence relates to the overall pattern of growth or decay in a geometric sequence.
The position of $a_4$ as the fourth term in a geometric sequence provides insight into the overall pattern of growth or decay. Since each term is a constant multiple of the previous term, the value of $a_4$ reflects the cumulative effect of the common ratio being applied three times to the first term, $a_1$. This allows you to analyze the rate of change in the sequence and make predictions about future terms, as well as understand the behavior of the sequence as a whole.
Analyze how the value of $a_4$ can be used to determine the explicit formula for the $n^{th}$ term in a geometric sequence.
Knowing the value of $a_4$ can be used to derive the explicit formula for the $n^{th}$ term in a geometric sequence, $a_n = a_1 \cdot r^{n-1}$. By rearranging the formula to solve for the common ratio, $r$, using the known values of $a_1$ and $a_4$, you can then substitute this value of $r$ back into the explicit formula to determine the expression for the $n^{th}$ term. This allows you to generalize the pattern observed in the sequence and apply the formula to calculate any term, not just $a_4$.
Related terms
Geometric Sequence: A sequence where each term is a constant multiple of the previous term, creating a pattern of growth or decay.
Common Ratio: The constant ratio between consecutive terms in a geometric sequence.
Explicit Formula: The formula that allows you to calculate any term in a geometric sequence based on the first term and the common ratio.