UK A-Level: Arithmetic and Geometric Series

Sequences and series are the mathematical backbone of patterns, from predicting loan repayments to modelling population growth. Mastering arithmetic and geometric progressions equips you with a powerful toolkit for describing linear change, exponential growth, and decay, which are central to A-Level Pure Mathematics and countless real-world applications.

Defining Sequences and Sigma Notation

A sequence is an ordered list of numbers. Each number in a sequence is called a term. Sequences can be defined explicitly by a formula for the $n$th term, $u_n$, in terms of $n$. For example, the sequence given by $u_n = 2n + 1$ generates 3, 5, 7, 9, ...

When we sum the terms of a sequence, we get a series. To express the sum of a series concisely, we use sigma notation ($\sum$). For instance, the sum of the first $n$ terms of a sequence is written as: $$S_n = \sum_{r=1}^{n} u_r$$ Here, $r$ is the index (starting at 1), $n$ is the upper limit, and $u_r$ is the rule for each term. Understanding sigma notation is essential for working formally with sums and is a frequent feature in exam questions.

Arithmetic Sequences and Series

An arithmetic sequence (or arithmetic progression, AP) is one where the difference between consecutive terms is constant. This constant is called the common difference, denoted by $d$.

If the first term is $a$, then the sequence is: $a$, $a+d$, $a+2d$, $a+3d$, ... This leads to the formula for the $n$th term: $$u_n = a + (n-1)d$$

To find the sum of the first $n$ terms, $S_n$, we use one of two standard formulas. You can derive them by writing the series forward and backward and adding: $$S_n = \frac{n}{2}[2a + (n-1)d] \quad \text{or} \quad S_n = \frac{n}{2}(a + l)$$ where $l$ is the last term ($u_n$). The second formula is particularly useful when you know the first and last term. For example, to find the sum of the first 20 positive integers (where $a=1$, $l=20$, $n=20$), you calculate $S_{20} = \frac{20}{2}(1+20) = 210$.

Geometric Sequences and Series

A geometric sequence (or geometric progression, GP) is one where the ratio between consecutive terms is constant. This constant is called the common ratio, denoted by $r$.

If the first term is $a$, the sequence is: $a$, $ar$, $ar^2$, $ar^3$, ... The formula for the $n$th term is: $$u_n = ar^{\,n-1}$$

The sum of the first $n$ terms of a geometric series has a distinct formula: $$S_n = \frac{a(1-r^{\,n})}{1-r} \quad \text{for} \quad r \neq 1$$ If $r=1$, the series is simply $a + a + a + ...$, so $S_n = na$. For example, to sum the first 5 terms of the GP 2, 6, 18,... (where $a=2$, $r=3$), you compute $S_5 = \frac{2(1-3^5)}{1-3} = \frac{2(1-243)}{-2} = 242$.

Infinite Geometric Series and Convergence

A powerful concept arises when we consider summing a geometric series forever. An infinite geometric series is the limit of $S_n$ as $n$ tends to infinity: $S_\infty = a + ar + ar^2 + ar^3 + ...$.

This sum only converges (approaches a finite limit) under a specific condition: when the common ratio $r$ satisfies $|r| < 1$. If $|r| \geq 1$, the terms do not diminish in magnitude and the sum diverges (goes to infinity or oscillates).

For a convergent infinite geometric series ($|r| < 1$), the sum to infinity is given by the elegant formula: $$S_\infty = \frac{a}{1-r}$$ This works because as $n \to \infty$, $r^{\,n} \to 0$. For instance, the series $12 + 6 + 3 + 1.5 + ...$ has $a=12$ and $r=\frac{1}{2}$. Since $|r|<1$, it converges, and $S_\infty = \frac{12}{1-\frac{1}{2}} = 24$.

Recurrence Relations and Modeling

Sequences are often defined by a recurrence relation, which expresses each term as a function of the previous term(s). A linear recurrence of the form $u_{n+1} = ku_n + c$ can model many real-world scenarios.

To work with these:

1. Identify if it defines an AP ($k=1$, $c=d$), a GP ($c=0$, $k=r$), or a more complex sequence.
2. You may be asked to find a fixed point (where $u_{n+1} = u_n$) or solve the relation to find an explicit formula for $u_n$.

Modeling is a key application. An arithmetic sequence models linear change, such as a weekly savings plan where you add a fixed amount. A geometric sequence models exponential growth (if $r>1$) or decay (if $0<r<1$), such as compound interest, radioactive decay, or the depreciation of a car's value. An infinite geometric series can model scenarios like the total vertical distance travelled by a bouncing ball that rebounds to a fraction of its previous height.

Common Pitfalls

1. Misidentifying the common difference or ratio: For an AP, ensure you subtract $u_{n} - u_{n-1}$ to find $d$. For a GP, divide $\frac{u_n}{u_{n-1}}$ to find $r$. A common error is to assume a sequence is geometric when the ratio between the first two terms doesn't hold for subsequent pairs.

Correction: Always check at least three consecutive terms to confirm the common difference or ratio is constant.

1. Applying the wrong sum formula: Using the infinite sum formula $S_\infty$ for a finite $n$, or using a geometric sum formula for an arithmetic series. A related error is using the sum formula $S_n = \frac{a(1-r^{\,n})}{1-r}$ without checking if $r=1$.

Correction: First, classify the series as AP or GP. For a GP, if $r=1$, use $S_n = na$. Only use $S_\infty = \frac{a}{1-r}$ if the question explicitly states "sum to infinity" and you have verified $|r| < 1$.

1. Errors with sigma notation: Misinterpreting the limits of summation. For $\sum_{r=3}^{7} (2r+1)$, the first term is when $r=3$, not $r=1$.

Correction: Write out the first few terms indicated by the sigma notation explicitly if you are unsure. Remember, $\sum_{r=1}^{n} r = \frac{n(n+1)}{2}$ is a useful standard result.

1. Overlooking convergence conditions: Attempting to calculate $S_\infty$ for a geometric series where $|r| \geq 1$.

Correction: Your very first step when seeing an infinite geometric series must be to state the condition for convergence and to verify that $|r| < 1$. If it isn't, state that the sum does not converge to a finite limit.

Summary

• The $n$th term of an Arithmetic Progression is $u_n = a + (n-1)d$; for a Geometric Progression, it is $u_n = ar^{\,n-1}$.
• The sum of the first $n$ terms of an AP is $S_n = \frac{n}{2}(2a+(n-1)d)$ or $\frac{n}{2}(a+l)$; for a GP, it is $S_n = \frac{a(1-r^{\,n})}{1-r}$ for $r \neq 1$.
• Sigma notation ($\sum$) provides a compact way to represent the sum of a series defined by a rule.
• An infinite geometric series converges to $S_\infty = \frac{a}{1-r}$ if and only if $|r| < 1$.
• Recurrence relations define sequences iteratively and are powerful tools for modeling real-world linear and exponential growth/decay scenarios.