Geometric Sequences & Series

• A geometric sequence is an ordered list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

• The first term of a geometric sequence is typically denoted by $a$. This is the starting value from which all subsequent terms are generated.

• The common ratio, denoted by $r$, is the constant factor by which each term is multiplied to obtain the next term. It can be found by dividing any term by its preceding term, i.e., $r = \frac{u_{n+1}}{u_n}$.

• The $n$-th term of a geometric sequence, $u_n$, represents the value at a specific position $n$ in the sequence. It is defined by the formula $u_n = ar^{n-1}$, where $a$ is the first term and $r$ is the common ratio.

• A geometric series is the sum of the terms of a geometric sequence. Instead of just listing the terms, a series involves adding them together, often represented using sigma notation, such as $\sum_{k=1}^{n} ar^{k-1}$.

• The common ratio $r$ is the fundamental principle governing the behavior of a geometric sequence. It dictates whether the sequence grows, shrinks, or alternates in sign.

• If $|r| > 1$, the terms of the sequence will increase in magnitude, leading to a diverging sequence. For example, if $r=2$, terms double with each step.

• If $0 < |r| < 1$, the terms of the sequence will decrease in magnitude, approaching zero. This leads to a converging sequence, where the sum of an infinite number of terms can be finite.

• If $r < 0$, the terms of the sequence will alternate in sign (positive, negative, positive, etc.). The sequence will still diverge if $|r| > 1$ or converge if $0 < |r| < 1$.

• If $r = 1$, all terms are equal to $a$, forming a constant sequence. If $r = -1$, terms alternate between $a$ and $-a$.

Finding the $n$-th Term

• The formula for the $n$-th term of a geometric sequence is $u_n = ar^{n-1}$. This formula allows for direct calculation of any term in the sequence given the first term $a$, the common ratio $r$, and the term's position $n$.

• To apply this, identify $a$ and $r$ from the given information. If $a$ and $r$ are not directly provided, they can often be deduced from two given terms by setting up and solving simultaneous equations.

• For instance, if $u_3 = 12$ and $u_5 = 48$, then $ar^2 = 12$ and $ar^4 = 48$. Dividing the second equation by the first yields $r^2 = 4$, so $r = \pm 2$. Substituting $r$ back allows for finding $a$.

Finding the Sum of the First $n$ Terms

• The sum of the first $n$ terms of a geometric series, denoted $S_n$, can be calculated using the formula:

$S_n = \frac{a(1-r^n)}{1-r}, \quad \text{for } r \neq 1$

• This formula is used when you need to find the total sum of a finite number of terms. It requires knowing the first term $a$, the common ratio $r$, and the number of terms $n$.

• An alternative form, $S_n = \frac{a(r^n-1)}{r-1}$, is often more convenient when $r > 1$ as it avoids negative denominators, though both formulas yield the same result.

• When solving problems involving $S_n$, you might be given $S_n$ and asked to find $a$, $r$, or $n$. This often involves algebraic manipulation, and finding $n$ may require the use of logarithms.

• The sum to infinity, $S_\infty$, represents the limiting value that the sum of a geometric series approaches as the number of terms $n$ tends towards infinity. This concept is only applicable to convergent series.

• A geometric series converges if and only if the absolute value of its common ratio is less than 1, i.e., $|r| < 1$. This condition means that the terms of the sequence must get progressively smaller and approach zero.

• If $|r| \ge 1$, the terms do not approach zero, and the sum of the series will either grow infinitely large or oscillate without settling on a finite value, meaning the series diverges and a sum to infinity does not exist.

• For a convergent geometric series (where $|r| < 1$), the sum to infinity is given by the formula:

$S_\infty = \frac{a}{1-r}, \quad \text{for } |r| < 1$

• This formula is powerful for calculating the total value of an infinitely long series, provided the convergence condition is met. It is widely used in areas like finance, physics, and probability.

• Understanding the differences between geometric and arithmetic series is crucial for selecting the correct formulas and problem-solving strategies.

• Arithmetic series involve a common difference ($d$) added to each term, leading to linear growth or decay. Geometric series involve a common ratio ($r$) multiplied by each term, resulting in exponential growth or decay.

• The formulas for the $n$-th term and the sum of $n$ terms are distinct for each type of series, reflecting their different underlying mathematical operations.

• Recognizing whether a problem describes an arithmetic or geometric progression is the first critical step in solving it. Look for keywords like 'added to each term' or 'multiplied by each term'.

• Incorrect Exponent for $n$-th Term: A common mistake is using $ar^n$ instead of $ar^{n-1}$ for the $n$-th term. Remember that the first term $a$ corresponds to $n=1$, so the exponent on $r$ should be $n-1$.

• Forgetting the Convergence Condition: When calculating the sum to infinity, students often forget to first check if $|r| < 1$. Calculating $S_\infty$ for a divergent series is a fundamental error.

• Mixing Up Formulas: Accidentally using an arithmetic series formula for a geometric series (or vice-versa) is a frequent source of error. Always identify the type of series first.

• Algebraic Errors with Logarithms: When solving for $n$ in $u_n = ar^{n-1}$ or $S_n = \frac{a(1-r^n)}{1-r}$, logarithms are often required. Errors in applying logarithm rules or calculator usage can lead to incorrect results.

• Sign Errors with Negative Ratios: When $r$ is negative, terms alternate in sign. Care must be taken with powers of $r$, especially $r^n$, to ensure the correct sign is maintained throughout calculations.

• Identify $a$ and $r$ First: For any geometric series problem, the very first step should be to clearly identify the first term ($a$) and the common ratio ($r$). These values are the foundation for all subsequent calculations.

• Check Convergence for $S_\infty$: Before attempting to calculate a sum to infinity, explicitly state and verify that the condition $|r| < 1$ is met. This demonstrates understanding of the underlying theory and can earn marks.

• Use Logarithms for $n$: If you need to find the number of terms $n$, especially when $n$ is in the exponent, be prepared to use logarithms. Ensure you are proficient with your calculator's logarithm functions.

• Simultaneous Equations: Problems often provide two terms of a sequence and ask for $a$ and $r$. Set up two equations using $u_n = ar^{n-1}$ and solve them simultaneously, typically by division to eliminate $a$.

• Contextualize Your Answer: After calculating a value, consider if it makes sense in the context of the problem. For example, if $r > 1$, $S_n$ should be growing rapidly, and $S_\infty$ should not exist. If $r$ is small, $S_\infty$ should be close to $a$.