Geometric sequences are all about multiplying by a constant factor. The common ratio determines how the sequence behaves, whether it grows, shrinks, or alternates. It's like a mathematical snowball effect, where each term builds on the previous one.

Understanding geometric sequences helps us model real-world situations with exponential growth or decay. We can use recursive and explicit formulas to find specific terms, and even calculate infinite sums when the sequence converges. It's a powerful tool for predicting patterns and trends.

Geometric Sequences

Common ratio in geometric sequences

• Constant factor multiplied by each term to get the next term in a geometric sequence
• Denoted as $r$ and calculated as $r = \frac{a_{n+1}}{a_n}$ for any $n \geq 1$ in the sequence $a_1, a_2, a_3, ...$
• Determines the behavior of the sequence:
  • $|r| > 1$ leads to exponential growth (powers of 2, 3, etc.)
  • $0 < |r| < 1$ results in exponential decay (halving, thirding, etc.)
  • $r = 1$ produces a constant sequence (repeating the same value)
  • $r = -1$ creates an alternating sequence between two values (1, -1, 1, -1, ...)
  • $r < -1$ or $-1 < r < 0$ generates an alternating sequence that grows or decays exponentially (doubling and negating, halving and negating, etc.)

Generation of geometric sequence terms

• Start with the first term $a_1$ and multiply by the common ratio $r$ repeatedly to generate subsequent terms
• Second term: $a_2 = a_1 \cdot r$
• Third term: $a_3 = a_2 \cdot r = a_1 \cdot r^2$
• Fourth term: $a_4 = a_3 \cdot r = a_1 \cdot r^3$
• General formula for the $n$-th term: $a_n = a_1 \cdot r^{n-1}$

Recursive formulas for geometric sequences

• Define each term based on the previous term using the common ratio $r$
• Recursive formula: $a_{n+1} = a_n \cdot r$ for $n \geq 1$
• Requires knowledge of the first term $a_1$ and common ratio $r$ to apply
• Enables extension of a geometric sequence by calculating the next term from the previous term and common ratio
• Example: Given $a_1 = 3$ and $r = 2$, find $a_2, a_3, a_4$ using the recursive formula
  1. $a_2 = a_1 \cdot r = 3 \cdot 2 = 6$
  2. $a_3 = a_2 \cdot r = 6 \cdot 2 = 12$
  3. $a_4 = a_3 \cdot r = 12 \cdot 2 = 24$

Explicit formulas for sequence terms

• Define the $n$-th term directly without relying on previous terms
• Explicit formula: $a_n = a_1 \cdot r^{n-1}$ for $n \geq 1$
• Requires knowledge of the first term $a_1$, common ratio $r$, and position $n$ of the desired term
• Allows finding specific terms without calculating all preceding terms
• Example: Find the 10th term in a geometric sequence with $a_1 = 2$ and $r = 3$ using the explicit formula
  • $a_{10} = 2 \cdot 3^{10-1} = 2 \cdot 3^9 = 39,366$

Sequence Behavior and Series

• Convergence and divergence describe the long-term behavior of geometric sequences
  • Convergence occurs when $|r| < 1$, causing the sequence to approach a limit
  • Divergence happens when $|r| \geq 1$, resulting in the sequence growing without bound or oscillating
• Partial sums represent the sum of a finite number of terms in a geometric sequence
• Infinite geometric series are the sum of all terms in an infinite geometric sequence
  • Only converge when $|r| < 1$, with the sum given by $S_{\infty} = \frac{a_1}{1-r}$
• Geometric sequences differ from arithmetic sequences, where terms increase by a constant difference rather than a constant ratio