Arithmetic and Geometric Sequences

Previous Knowledge Required: Sequences and limits (S6).

A sequence $\{u_n\}_{n=1}^\infty$ in an ordered list of items. We use the shorthand notation $u_n$ to refer to the $n$ -th term of the sequence, so that the symbol $\{u_n\}_{n=1}^\infty$ refers to a collection:

$\{u_n\}_{n=1}^\infty=\{u_1,u_2,u_3,...\}$

Table of Contents

Quick Reminder from S6

Sometimes these sequences will be defined explicitly with some formula (possibly depending on $n$ ) and sometimes recursively (depending on previous terms of the sequence). We can use any letter $u_n, a_n, b_n,...$ to refer to a specific sequence, and the subscript $n$ refers to the term in that place of the sequence. That is: $a_1$ refers to the first term of the sequence $a_n$ and so on. Some sequences may even start from a 0-th term, so that you may also see the notation $\{u_n\}_{n=0}^\infty$ to indicate the very first term in the sequence is $u_0$ .

Example: In an explicitly defined sequence, any term $u_n$ can be calculated directly using its position index ( $n$ ), without needing to know the preceding terms. Unlike arithmetic or geometric sequences that rely on constant addition or multiplication, explicit sequences can follow any mathematical function.

Consider the sequence defined explicitly by the non-linear rule:

$u_n = n^2 - n + 2 \quad \text{for } n \ge 1$

Using this explicit rule, the first 8 terms are evaluated directly by substituting the position values $1$ through $8$ for $n$ :

Position ( $n$ )	Direct Substitution Steps	Term Value ( $u_n$ )
1	$(1)^2 - 1 + 2 = 1 - 1 + 2$	2
2	$(2)^2 - 2 + 2 = 4 - 2 + 2$	4
3	$(3)^2 - 3 + 2 = 9 - 3 + 2$	8
4	$(4)^2 - 4 + 2 = 16 - 4 + 2$	14
5	$(5)^2 - 5 + 2 = 25 - 5 + 2$	22
6	$(6)^2 - 6 + 2 = 36 - 6 + 2$	32
7	$(7)^2 - 7 + 2 = 49 - 7 + 2$	44
8	$(8)^2 - 8 + 2 = 64 - 8 + 2$	58

Therefore, the first 8 terms of this explicit sequence are: 2, 4, 8, 14, 22, 32, 44, 58 and we can replace any $n$ to find more and more terms.

We mentioned it already, but sequences can also be defined using one or more previous terms in the sequence. These are referred to as recursive sequences.

Example: The Fibonacci sequence is a classic example of a recursive sequence. Each term is generated by performing operations on the preceding terms. Specifically, each new number in this pattern is formed by calculating the sum of the two numbers immediately before it.

The sequence is formally defined by the mathematical rule:

$F_1 = 1$

$F_2 = 1$

$F_n = F_{n-1} + F_{n-2} \quad \text{for } n \ge 3$

Following this recursive blueprint, the first 8 terms are computed as follows:

Term Symbol	Recursive Calculation Steps	Resulting Value
$F_1$	Given Initial Baseline	1
$F_2$	Given Initial Baseline	1
$F_3$	$F_2 + F_1 = 1 + 1$	2
$F_4$	$F_3 + F_2 = 2 + 1$	3
$F_5$	$F_4 + F_3 = 3 + 2$	5
$F_6$	$F_5 + F_4 = 5 + 3$	8
$F_7$	$F_6 + F_5 = 8 + 5$	13
$F_8$	$F_7 + F_6 = 13 + 8$	21

Therefore, the first 8 terms of the standard Fibonacci sequence are: 1, 1, 2, 3, 5, 8, 13, 21 and it continues forever.

You have also learned to do some basic limit calculation, though you will also expand on this topic in S7 when you begin the analysis unit.

Arithmetic Sequences

Definition (Arithmetic Sequence): An arithmetic sequence is a sequence of numbers where the difference between any consecutive terms is a constant value. This constant value is known as the common difference ( $d$ ).

The sequence can be defined recursively as:

$u_{n} = u_{n-1} + d \quad \text{for } n \ge 2$

Alternatively, it can be expressed explicitly using its initial term $u_1$ as:

$u_n = u_1 + (n - 1)d \quad \text{for } n \ge 1$

This alternative definition can be concluded very easily by looking at the terms of the sequence. If we add $d$ to go from $u_1$ to $u_2$ and then $d$ again to go from $u_2$ to $u_3$ then we go from $u_1$ to $u_3$ by adding $d$ twice. This is what the 2nd formula is basically saying. To go from $u_1$ to $u_n$ we must add $d$ a total of $(n-1)$ times.

Example (Arithmetic Sequence): Consider an arithmetic sequence defined by its first term $u_1 = 5$ and a common difference $d = 3$ . Find the first 5 terms of the sequence, and compute the value of the $20\text{th}$ term ( $u_{20}$ ).

We can also be asked to add terms in a sequence, and in the case of arithmetic sequences, we have a rather quick way of doing so.

Claim (Sum of terms in an Arithmetic Sequence): The sum of the first $n$ terms of an arithmetic sequence, denoted as $S_n = u_1 + u_2 + \dots + u_n$ , can be evaluated using the first term $u_1$ and the common difference $d$ .

$S_n=\sum_{i=1}^n a_i=a_1+a_2+...+a_n = \frac{n\cdot(u_1+u_n)}{2}$

Example (Finite Arithmetic Sum): An infrastructure project expands its solar grid capacity incrementally each month. In the first month, they install $14$ solar panels. By the final month of the phase, month $24$ , their operational expansion rate reaches exactly $106$ panels installed that month. Calculate the total number of solar panels installed over this $24$ -month period.

You could also be asked to find terms or the rule that defines an arithmetic sequences and use some of these formulas to understand the connection.

Example: For a given arithmetic sequence, the third term is $13$ ( $u_3 = 13$ ) and the fifth term is $21$ ( $u_5 = 21$ ). Find the first term ( $u_1$ ), the common difference ( $d$ ), and state the general explicit rule for this sequence.

Geometric Sequences

Definition (Geometric Sequence): A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero constant value. This constant value is known as the common ratio ( $r$ or $q$ depending on the source).

The sequence can be defined recursively as:

$u_{n} = r \cdot u_{n-1} \quad \text{for } n \ge 2$

Alternatively, it can be expressed explicitly using its initial term $u_1$ as:

$u_n = u_1 \cdot r^{n-1} \quad \text{for } n \ge 1$

Example (Geometric Sequence): Consider a geometric sequence defined by its first term $u_1 = 6$ and a common ratio $r=2$ . Find the first 5 terms of the sequence, and compute the value of the $10\text{th}$ term ( $u_{10}$ ).

Claim (Sum of terms in a Geometric Sequence): The sum of the first $n$ terms of a geometric sequence with common ratio $r \neq 1$ is given by $S_n$ . If $|r| < 1$ , the infinite sum $S_\infty$ converges to a fixed limit.

$S_n =\sum_{i=1}^n= u_1 \frac{1 - r^n}{1 - r} \quad \text{and} \quad S_\infty = \sum_{i=1}^\infty u_i=\frac{u_1}{1 - r} \,\, \text{for } |r| < 1$

Another way of showing this claim is by using the identity:

$(1+x+x^2+...+x^{n-1})(x-1)=x^n-1\implies 1+x+x^2+...+x^{n-1}=\frac{x^n-1}{x-1}$

The left-hand side is a geometric sequence, starting from $1$ and always being multiplied by $x$ (or $r$ if you will). The difference between this and the formula above is a factor of $u_1$ that would otherwise be multiplying every term (when the sequence does not start with 1).

This yields an explanation to a very interesting phenomenon: we can have an infinite sum, where every term is positive, but the sum is finite. The next example will show this in detail.

Example (Finite and Infinite Geometric Sequence Sums): Consider the geometric sequence defined explicitly by the rule $u_n = \left(\frac{1}{2}\right)^{n-1}$ for $n \ge 1$ . The first few terms of this sequence are generated as follows:

$u_1 = \left(\frac{1}{2}\right)^0 = 1$
$u_2 = \left(\frac{1}{2}\right)^1 = \frac{1}{2}$
$u_3 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$
$u_4 = \left(\frac{1}{2}\right)^3 = \frac{1}{8}$

For this sequence, evaluate the following metrics:

Find the sum of the first $8$ terms of this sequence: $S_8 = u_1 + u_2 + \dots + u_8$ .
Determine the exact limit of the sum of all terms ( $S_\infty$ ) as the number of terms approaches infinity.

Another way of thinking about this example is the following: Imagine you have a glass of water you want to fill up. At first you fill half of it with water. Then you fill half of what is left (i.e.: thus far you have 3/4 of the glass filled). If you continue like this, you will end up with the glass full, but not overflowing. This is because you are always filling up half of what is left, so at any finite point you will always have a bit left from which only half is filled.

Example: A finite geometric sequence has $10$ terms and a common ratio $r = 2$ . Given that the sum of all terms with an odd index ( $u_1 + u_3 + u_5 + u_7 + u_9$ ) is equal to $1364$ , determine the first term ( $u_1$ ) of the sequence.

Combining Sequences

Sequences do not necessarily depend exclusively on $n$ , but rather could be defined using other sequences. Here is an example of such a sequence.

Example: Let $(u_n)$ be defined by

$\begin{cases}u_0 = 1,\\u_{n+1} = 3u_n + 2 \quad (n \ge 0)\end{cases}$

Define a second sequence $(v_n)$ by

$v_n = u_n + 1.$

Show that $(v_n)$ is a geometric sequence.
Determine its common ratio and first term.
Find an explicit formula for $v_n$ .
Deduce an explicit formula for $u_n$ .

Exercises

Consider an arithmetic sequence defined by its first term $u_1 = 7$ and a common difference $d = 4$ . Solve the following sub-problems:

1. Write down the first $4$ terms of this sequence.
2. State the explicit rule for the general term $u_n$ of the sequence.
3. Find the value of the $15\text{th}$ term ( $u_{15}$ ).
4. Calculate the sum of the first $30$ terms of this sequence ( $S_{30}$ ).

Consider a geometric sequence defined by its first term $u_1 = 3$ and a common ratio $r = 2$ . Solve the following sub-problems:

1. Write down the first $4$ terms of this sequence.
2. State the explicit rule for the general term $u_n$ of the sequence.
3. Find the value of the $8\text{th}$ term ( $u_{8}$ ).
4. Calculate the sum of the first $10$ terms of this sequence ( $S_{10}$ ).

Consider a sequence defined explicitly by the rule $u_n = 5n - 3$ for $n \ge 1$ . Solve the following sub-problems:

1. Write down the first $3$ terms of this sequence to observe the initial pattern.
2. Write an algebraic expression for the next consecutive term, $u_{n+1}$ , in terms of $n$ .
3. Prove that the sequence is arithmetic by expanding and simplifying the difference expression $u_{n+1} - u_n$ .
4. State the first term ( $u_1$ ) and the common difference ( $d$ ) of this sequence based on your proof.

Consider a sequence defined explicitly by the rule $u_n = 4 \cdot 3^{n-1}$ for $n \ge 1$ . Solve the following sub-problems:

1. Write down the first $3$ terms of this sequence to observe the initial pattern.
2. Write an algebraic expression for the next consecutive term, $u_{n+1}$ , in terms of $n$ .
3. Prove that the sequence is geometric by simplifying the quotient expression $\frac{u_{n+1}}{u_n}$ using index laws.
4. State the first term ( $u_1$ ) and the common ratio ( $r$ ) of this sequence based on your proof.

Consider a geometric sequence defined explicitly by the rule $u_n = 24 \cdot \left(\frac{1}{3}\right)^{n-1}$ for $n \ge 1$ . Solve the following sub-problems:

1. Write down the first $3$ terms of this sequence to observe the initial scaling.
2. Identify the first term ( $u_1$ ) and the common ratio ( $r$ ).
3. State why this geometric sequence is guaranteed to converge as the number of terms approaches infinity.
4. Use the infinite sum formula to calculate the sum of all terms.

Let $(u_n)$ be defined by the recurrence relation:

$\begin{cases}u_0 = 3, \\u_{n+1} = 2u_n + 5 \quad (n \ge 0)\end{cases}$

Define an auxiliary sequence $(v_n)$ by the translation rule:

$v_n = u_n + 5.$

1. Show algebraically that $(v_n)$ forms a geometric sequence.
2. Determine the common ratio ( $r$ ) and the first term ( $v_0$ ) of this sequence framework.
3. Find an explicit formula for $v_n$ in terms of $n$ .
4. Deduce an explicit formula for the original sequence $u_n$ in terms of $n$ .

Let $(u_n)$ be defined by the recurrence relation:

$\begin{cases}u_0 = 4, \\ u_{n+1} = 4u_n - 9 \quad (n \ge 0)\end{cases}$

Define an auxiliary sequence $(v_n)$ by the translation rule:

$v_n = u_n - 3.$

1. Show algebraically that $(v_n)$ forms a geometric sequence.
2. Determine the common ratio ( $r$ ) and the first term ( $v_0$ ) of this sequence framework.
3. Find an explicit formula for $v_n$ in terms of $n$ .
4. Deduce an explicit formula for the original sequence $u_n$ in terms of $n$ .

Let $(u_n)$ be defined by the recurrence relation:

$\begin{cases}u_0 = 1,\\ u_{n+1} = 5u_n - 8 \quad (n \ge 0)\end{cases}$

Define an auxiliary sequence $(v_n)$ by the translation rule:

$v_n = u_n - 2.$

1. Show algebraically that $(v_n)$ forms a geometric sequence.
2. Determine the common ratio ( $r$ ) and the first term ( $v_0$ ) of this sequence framework.
3. Find an explicit formula for $v_n$ in terms of $n$ .
4. Deduce an explicit formula for the original sequence $u_n$ in terms of $n$ .