Sequences are a useful tool to help us ease into Calculus. We can do a lot of things with sequences, so hopefully we will be able to understand basic concepts with those. We will be able to re-use those concepts when we will want to study functions.

Basic concepts

Definition and examples

A sequence is a list of real numbers. Each number is associated with a positive integer. For example,

• If you have funds on a saving account and you let the interest rate compound, you can denote each element of the sequence as the balance at the end of the nth year.
• Geometrically, if you have a circle of radius 1 and you define a regular polygon of $n$ sides inscribed in this circle, you can study the sequence of the length of all sides for each $n$. (Let’s say n begins at $3$ otherwise it doesn’t make too much sense).

Expressions

Most of the sequences can be expressed in two different ways:

• Either recursively, which means that the value at step $n$ is expressed by a function of the previous steps values. Most often it is defined as a function of the previous value. Sometimes it is defined with more than one previous value.
• Or directly, which means that you have a direct formula to compute the value at step $n$.

For example, in our compounded interest example, we can express the sequence in both ways:

• recursively, with $A_n = A_{n-1}*{1+r}$
• directly, with $A_n = A_0*(1+r)^n$ (Note: how would you prove that both expressions are equivalent?)

It is better (if possible) to get a direct expression for the sequence, as it would make things easier to study their properties later on. But, hey, we’re just doing our best here. 😅

Exercises

1. Can you think about sequences that you encounter in your daily life? Think about sequences which can be indexed with positive integers. (Number of weeks? Number of years?) Maybe in daily life, we struggle to think about sequences which can be infinite, but in mathematics we can just think about continuing the pattern forever.
2. Write the first 5 elements of the sequence defined by $a_n = 2a_{n-1}$ with $a_0 = 1$.
3. Write the first 5 elements of the sequence defined by $a_n = a_{n-1} + 3$ with $a_0 = 1$.
4. Compute the first 10 elements of the sequence defined by $a_n = a_{n-1} + a_{n-2}$ with $a_0 = 1$ and $a_1 = 1$. (This is the famous Fibonacci sequence.)