IB AA: Sequences and Series

Sequences and series form the mathematical backbone for modeling growth, decay, and recurring sums, from calculating loan payments to analyzing fractal patterns. In the IB Analysis & Approaches (AA) course, you move beyond simple pattern recognition to a rigorous algebraic treatment, equipping you with tools essential for higher mathematics and real-world financial applications. Mastering this topic is crucial for exams and provides a foundation for calculus and discrete mathematics.

Core Concepts: From Patterns to Formulas

A sequence is an ordered list of numbers, each called a term. The two most fundamental types are arithmetic and geometric sequences, defined by how you get from one term to the next.

An arithmetic sequence is one where the difference between consecutive terms is constant. This constant is called the common difference, denoted by $d$. If you know the first term, $u_1$, the $n$th term is given by: $$u_n=u_1+(n-1)d$$ For example, the sequence 2, 5, 8, 11,... is arithmetic with $d=3$. The 10th term would be $u_{10}=2+(10-1)\times 3=29$.

A geometric sequence is one where the ratio between consecutive terms is constant. This constant is called the common ratio, denoted by $r$. The $n$th term formula is: $$u_n=u_1\cdot r^{n-1}$$ Consider the sequence 3, 6, 12, 24,... Here, $r=2$. The 6th term is $u_6=3\times 2^5=96$.

Sigma Notation and Finite Series

The sum of a sequence is called a series. Sigma notation ($\sum$) provides a concise way to represent the sum of a series. For instance, ${\sum }_{k=1}^5(2k+1)$ means to sum the expression $(2k+1)$ for $k=1$ to $k=5$, resulting in $3+5+7+9+11=35$.

More importantly, we have formulas for the sum of the first $n$ terms of arithmetic and geometric sequences, known as $S_n$.

For an arithmetic series, the sum is: $$S_n=\frac{n}{2}(2u_1+(n-1)d)\text{or}{S}_n=\frac{n}{2}(u_1+u_n)$$ The second form is often quicker if you know the last term.

For a finite geometric series, the sum is: $$S_n=\frac{u_1(1-r^n)}{1-r},r\ne 1$$ For example, to sum the first 4 terms of the geometric sequence 5, -10, 20, -40,... you have $u_1=5$, $r=-2$, and $n=4$. Thus, $S_4=\frac{5(1-(-2{)}^4)}{1-(-2)}=\frac{5(1-16)}{3}=\frac{-75}{3}=-25$.

Infinite Geometric Series and Convergence

Not all series have a finite sum. However, an infinite geometric series ($S_{\infty }$) converges to a finite value if and only if the absolute value of the common ratio is less than one: $|r|<1$. When this condition is met, the sum to infinity is given by: $$S_{\infty }=\frac{u_1}{1-r},|r|<1$$ This result is powerful. Consider the series $18+6+2+\frac{2}{3}+...$. Here, $u_1=18$ and $r=\frac{1}{3}$. Since $|r|<1$, the series converges: $S_{\infty }=\frac{18}{1-\frac{1}{3}}=\frac{18}{\frac{2}{3}}=27$. If $|r|\ge 1$, the series diverges, meaning its sum grows without bound or oscillates.

Applications to Financial Mathematics

Sequences and series directly model core financial concepts. Compound interest is a geometric sequence. If a principal $P$ is invested at an annual interest rate $r$, compounded annually, the value after $n$ years is $A_n=P(1+i{)}^n$, where $i$ is the interest rate per compounding period.

More nuanced are annuities. An annuity is a series of equal regular payments or investments. For instance, if you invest a fixed amount $PMT$ at the end of each year into an account with an annual interest rate $i$, the future value $FV$ after $n$ years is the sum of a geometric series: $$FV=PMT\cdot \frac{(1+i{)}^n-1}{i}$$ This formula is derived directly from the sum of a geometric series where the first term is $PMT(1+i{)}^{n-1}$. Similarly, loans and mortgages use present value calculations involving geometric series.

Proof by Mathematical Induction for Series

The IB syllabus requires you to prove certain formulas, particularly for sums of series, using proof by induction. This is a two-step process used to prove a statement $P(n)$ is true for all positive integers $n$. For a series formula, the steps are:

1. Base Case: Verify the formula holds for the initial value, usually $n=1$.
2. Inductive Step: Assume the formula is true for $n=k$ (the inductive hypothesis). Then, using this assumption, prove it must also be true for $n=k+1$.

By completing these steps, you logically demonstrate the truth of the formula for all $n$. This method is frequently applied to prove the standard formulas for the sum of the first $n$ positive integers (${\sum }_{k=1}^{n}k=\frac{n(n+1)}{2}$) or the sum of squares.

Common Pitfalls

1. Confusing Sequence and Series: A sequence is a list; a series is the sum of that list. Using a term formula when a sum formula is needed (or vice versa) is a frequent exam error. Always read the question carefully: is it asking for a term ($u_n$) or a sum ($S_n$)?
2. Misapplying the Infinite Sum Formula: The formula $S_{\infty }=\frac{u_1}{1-r}$ only applies if $|r|<1$. Always check this condition first. Using it for a series where $r=2$ will give a nonsensical answer.
3. Arithmetic vs. Geometric Confusion: An arithmetic sequence has a constant difference (add/subtract). A geometric sequence has a constant ratio (multiply/divide). Mistaking one for the other will lead to using the wrong $n$th term and sum formulas. Identify the pattern correctly by checking $u_n-u_{n-1}$ and $u_n/u_{n-1}$.
4. Errors in Sigma Notation: Pay close attention to the limits of summation. The expression ${\sum }_{k=0}^{n}a{r}^k$ has $n+1$ terms (from $k=0$ to $n$ inclusive), while ${\sum }_{k=1}^{n}a{r}^{k-1}$ has $n$ terms. The underlying series may be identical, but the formulas you apply must match the starting index.

Summary

• Arithmetic sequences are defined by a common difference $d$, with $u_n=u_1+(n-1)d$ and sum $S_n=\frac{n}{2}(2u_1+(n-1)d)$.
• Geometric sequences are defined by a common ratio $r$, with $u_n=u_1{r}^{n-1}$ and finite sum $S_n=\frac{u_1(1-r^n)}{1-r}$ for $r\ne 1$.
• An infinite geometric series converges to $S_{\infty }=\frac{u_1}{1-r}$ only when $|r|<1$.
• These concepts have direct financial applications, modeling compound interest ($A=P(1+i{)}^n$) and annuities.
• Proof by induction is a required method for formally proving formulas for the sum of series, involving a base case and an inductive step.