Find the Explicit Formula - Algebra-1

1. Fundamental Concepts

Geometric Sequence: A sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio (r).
Explicit Formula: The n-th term of a geometric sequence can be calculated directly using the formula:\(a_n = a_1 \cdot r^{n-1}\) where \(a_1\) is the first term, r is the common ratio, and n is the term number.
Identification Criterion: A sequence is geometric if the ratio of any consecutive terms (latter term ÷ former term) is constant; otherwise, it is not.

Calculation of Common Ratio: \(r = \frac{a_{n+1}}{a_n}\) (for any \(n \geq 1\)).
Alternating Signs: When the common ratio r is negative, the signs of the sequence terms alternate (e.g., with \(r = -2\), the sequence might be \(2, -4, 8, -16, \dots\)).
Special Cases:
- When \(r = 1\), the sequence is a constant sequence (e.g., \(1, 1, 1, \dots\)), and the explicit formula simplifies to \(a_n = a_1\).
- If a sequence contains 0 with non-zero subsequent terms, it is not a geometric sequence (e.g., \(4, 0, 0, \dots\) because the common ratio cannot be calculated).

Question: Is the sequence \(1, 1, 1, \dots\) geometric? If yes, find \(a_8\).
Solution:
- The common ratio \(r = \frac{1}{1} = 1\), so it is a geometric sequence.
- Explicit formula: \(a_n = 1 \cdot 1^{n-1} = 1\).
- \(a_8 = 1\).

Question: Is the sequence \(3, -3, 3, -3, \dots\) geometric? If yes, find \(a_6\).
Solution:
- The common ratio \(r = \frac{-3}{3} = -1\), so it is a geometric sequence.
- Explicit formula: \(a_n = 3 \cdot (-1)^{n-1}\).
- \(a_6 = 3 \cdot (-1)^5 = 3 \cdot (-1) = -3\).

Question: Is the sequence \(-4, 8, -16, 32, \dots\) geometric? If yes, find \(a_6\).
Solution:
- The common ratio \(r = \frac{8}{-4} = -2\), so it is a geometric sequence.
- Explicit formula: \(a_n = -4 \cdot (-2)^{n-1}\).
- \(a_6 = -4 \cdot (-2)^5 = -4 \cdot (-32) = 128\).

Identifying a Geometric Sequence: Calculate the ratio of consecutive terms. If all ratios are equal, the sequence is geometric, and this ratio is the common ratio r. Example: In \(2, 10, 50, \dots\), \(\frac{10}{2} = 5\) and \(\frac{50}{10} = 5\), so \(r = 5\), and it is a geometric sequence.
Handling Sequences with Alternating Signs: If the common ratio r is negative, the terms alternate in sign. Directly substitute into the explicit formula, noting the sign rule for negative exponents: negative numbers raised to odd powers are negative, and to even powers are positive.
Calculating a Specific Term Using the Explicit Formula: Once \(a_1\) (the first term) and r (the common ratio) are determined, substitute them into \(a_n = a_1 \cdot r^{n-1}\) along with the term number n to find the desired term. Example: In the sequence \(96, 24, 6, \dots\), \(a_1 = 96\) and \(r = \frac{24}{96} = \frac{1}{4}\). Then \(a_6 = 96 \cdot \left( \frac{1}{4} \right)^5 = 96 \cdot \frac{1}{1024} = \frac{3}{32}\).