Arithmetic and Geometric Sequences in Discrete Mathematics

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In Discrete Mathematics, we use the concept of sequences to understand patterns and predict the subsequent terms in ordered lists of numbers. Among the most common types are arithmetic and geometric sequences. These two sequence types form the basics to understand and predict numbers that increase either additively or multiplicatively.

In this chapter, we will explain the basics of Arithmetic and Geometric Sequences and provide a set of examples to highlight their practical uses.

What is a Sequence?

A sequence is an ordered list of numbers. Each number in the sequence is called a term. Arithmetic and geometric sequences differ in how each term is generated from the previous one. Arithmetic sequences add a fixed number (known as the common difference), while geometric sequences multiply by a constant (called the common ratio).

Arithmetic Sequences

In the arithmetic sequence it is one in which each term is generated by adding a constant. We know this as the common difference (denoted as d). This is added to the previous term. For example, in the sequence 2, 5, 8, 11, … each term increases by 3. This consistent difference is used to characterize the sequence as arithmetic.

Recursive Definition

In an arithmetic sequence, we can understand that each term is based on the previous term and the common difference. The recursive formula is −

$$\mathrm{a_n \:=\: a_{n-1} \:+\: d}$$

where a₀ is the initial term, and d is the common difference. This formula defines each term as the previous term plus d.

Closed Formula

The idea of closed formula is used to describe a sequence other than recurrence relations. A closed formula for an arithmetic sequence is used to calculate any term directly, without knowing the previous term.

For an arithmetic sequence with first term a₀ and common difference d, the closed formula is −

$$\mathrm{a_n \:=\: a_0 \:+\: n \:\cdot\: d}$$

This formula shows the n-th term is the first term plus d multiplied by n, the term position.

Let us see some examples for a better understanding:

Example 1 − Let us calculate the terms in the sequence 3, 7, 11, 15, where the first term a₀ = 3 and d = 4.

Recursive definition −

$$\mathrm{a_n \:=\: a_{n-1} \:+\: 4}$$

Closed formula −

$$\mathrm{a_n \:=\: 3 \:+\: 4n}$$

Using the closed formula, we can find the fifth term that is a₄ −

$$\mathrm{a_4 \:=\: 3 \:+\: 4 \:\cdot\: 4}$$

So, the fifth term is 19.

Example 2 − For another sequence let us say 100, 90, 80, 70, … we see that each term decreases by 10.

• Recursive formula − a_n = a_n−1 – 10 with a₀ = 100.
• Closed formula − a_n = 100 − 10n.

To find the seventh term it is a₆ −

$$\mathrm{a_6 \:=\: 100 \:-\: 10 \:\cdot\: 6 \:=\: 40}$$

This gives the seventh term as 40.

Geometric Sequences

The next type of sequences are geometric sequences. These differs from an arithmetic sequence in that each term is generated by multiplying the previous term by a constant, known as the common ratio (denoted as r). An example is the sequence 3, 6, 12, 24, … where each term is twice the previous term.

Recursive Definition

The recursive formula for a geometric sequence shows multiplication with a fixed ratio −

$$\mathrm{a_n \:=\: a_{n-1} \:\cdot\: r}$$

where a0 is the initial term, and r is the common ratio.

Closed Formula

The closed formula for geometric sequences provides a way to find any term in the sequence without knowing the previous terms.

For a geometric sequence with initial term a₀ and common ratio r, the closed formula is −

$$\mathrm{a_n \:=\: a_0 \:\cdot\: r^n}$$

This formula expresses the n-th term as the initial term multiplied by r raised to the power of n.

Let us now check a couple of examples of geometric sequences.

Example 1 − Let us see the sequence 2, 6, 18, 54, … where a₀ = 2 and r = 3

Recursive definition −

$$\mathrm{a_n \:=\: a_{n-1} \:\cdot\: 3}$$

Closed formula

$$\mathrm{a_n \:=\: 2 \:\cdot\: 3^n}$$

To find the fourth term a₃ −

$$\mathrm{a_3 \:=\: 2 \:\cdot\: 3^3 \:=\: 2 \:\cdot\: 27 \:=\: 54}$$

So, the fourth term is 54.

Example 2 − For the sequence 81, 27, 9, 3, … each term is one-third of the previous term. This  means the common ratio is 1/3.

Recursive formula −

$$\mathrm{a_n \:=\: a_{n-1} \:\cdot\: \frac{1}{3} \quad \text{with} \quad a_0 \:=\: 81}$$

Closed formula −

$$\mathrm{a_n \:=\: 81 \:\cdot\: \left( \frac{1}{3} \right)^n}$$

Calculating the fifth term a₄ −

$$\mathrm{a_4 \:=\: 81 \:\cdot\: \left( \frac{1}{3} \right)^4 \:=\: 81 \:\cdot\: \frac{1}{81} \:=\: 1}$$

The fifth term is 1.

Real-World Applications of Sequences

Both sequences, the arithmetic and geometric are used in several cases everywhere, from finance to engineering. Here are some practical applications.

Arithmetic Sequence: Salary Growth

Consider a company gives its employees a yearly salary increase of Rs. 50000. If an employee starts with a salary of Rs. 500000, we can model this with an arithmetic sequence −

Initial term, $\mathrm{a_0 \:=\: 500000}$

Common difference, $\mathrm{d \:=\: 50000}$

Using the closed formula a_n = 500000 + 50000 ⋅ n, we can find the employee's salary after 5 years −

$$\mathrm{a_5\:=\:500000\:+\:50000 \:\cdot\: 5 \:=\: 750000}$$

So, after five years, the salary would be Rs. 750000.

Geometric Sequence: Population Growth

Let us consider a town has a population of 1000 people, and it grows by 5% each year. This growth forms a geometric sequence −

Initial population $\mathrm{a_0\:=\: 1000}$

Common ratio, $\mathrm{r \:=\: 1.05}$

To predict the population in 10 years, we use the closed formula −

$$\mathrm{a_10 \:=\: 1000 \:\cdot\: (1.05)^{10}}$$

Calculating this, we find the population would be about 1629 in ten years.

Geometric Sequence: Compound Interest

Another great example for geometric series is compound interest in savings accounts. Suppose we invest Rs.5000 with an interest rate of 4% compounded annually. This would follow a geometric sequence −

Initial amount, $\mathrm{a_0 \:=\: 5000}$

Common ratio, $\mathrm{r \:=\: 1.04}$

The formula to find the amount after 6 years is −

$$\mathrm{a_6 \:=\: 5000 \:\cdot\: (1.04)^6}$$

The amount would grow to about Rs. 6326 after 6 years.

Conclusion

In this chapter, we explained the concept of arithmetic and geometric sequences in discrete mathematics. Arithmetic sequences involve adding a constant difference, while geometric sequences rely on multiplying a common ratio. For each type, we understood both recursive and closed formulas.

We also highlighted some real-world applications of these sequences, from salary increases to population growth and compound interest.