Explicit Formulas

In a sequence to find any term without knowing the previous term, we use the explicit formula. It is a formula used to find the nth term of a sequence based on its position. Let’s learn the explicit formula for different types of sequences:

To find the nth term of a sequence, we use an explicit formula. The sequence can be arithmetic, geometric, or harmonic.

nth term of an arithmetic sequence -  a_n = a + (n - 1)d, where
a is the first term

n is the position of the term in the sequence, 

and d is the common difference. Steps used for finding explicit formulas are -

Step 1: First find the first term and the common difference of the sequence

Step 2: Substitute the value of a, n, and d in the explicit formula, a_n = a + (n - 1)d

Step 3: Simplify the formula to find the nth term

To find the 7th term of the sequence 7, 14, 21, 28, … 

In the given sequence, a = 7 and d = 7(14 - 7 = 7)

The nth term of an arithmetic sequence is: a_n = a + (n - 1)d

So, the 7th term is: a7 = 7 + (7 - 1)7

= 7+ (6 × 7)

= 7 + 42 = 49

Therefore, the 7th term of the sequence is 49.

In an arithmetic sequence, the difference between consecutive terms is constant and is called the common difference (d). To find the nth term of an arithmetic sequence, we use the explicit formula: an = a + (n - 1)d. 
Where, 

• a_n is the nth term

• a is the first term

• d is the common difference

For example, for the arithmetic sequence 2, 5, 8, 11, 14, …, finding the explicit formula
 
Here, a = 2

d = 3

The explicit formula of an arithmetic sequence is: an = a + (n - 1)d
Substituting the value of a and d:

a_n = 2 + (n - 1)3

= 2 + 3n - 3 

a_n = 3n - 1 

Find the 25th term of the sequence. 

a₂₅ = 3 × 25 - 1 

= 75 - 1

= 74

Therefore, the 25th term of the sequence is 74. 

The geometric sequence is any sequence where the ratio of any two consecutive terms is the same. The ratio is known as the common ratio (r). The general form of a geometric sequence can be represented as a, ar, ar², ar³, … ar^{n - 1}. For the geometric sequence, the explicit formula is an = ar^{n - 1}. 

Where, 

• a is the first term

• a_n is the nth term 

• r is the common ratio

For example, for the sequence: 1, 2, 4, 8, …, finding the explicit formula

Here, a = 1

r = 2

The explicit formula for geometric sequences is: a_n = ar^{n - 1}
Substituting the value of a and r:

a_n = 1 × 2^{n - 1}
= 2^{n - 1}

Finding the 5th term:

a_{5 =} 2^{(5 - 1)}

= 2⁴ = 16

So, the 5th term is 16 

A harmonic sequence is a type of sequence where the reciprocals of the terms form an arithmetic sequence. For instance, the harmonic sequence of 2, 4, 6, 8, … is 1/2, 1/4, 1/6, 1/8, …. The general form of a harmonic sequence can be represented as 1/a, 1/(a +d), 1/(a + 2d), …, 1/(a + (n - 1)d). For a harmonic sequence, the explicit formula: a_n = 1/(a + (n - 1)d)

• Where a is the first term

• a_n is the nth term

• d is the common difference

For example, find the explicit formula for the harmonic sequence: 1/3, 1/6, 1/9, 1/12. 

We take the reciprocals of the terms: 3, 6, 9, 12, …, to find a and d. 
Here, a = 3

d = 3

a_n = 1/(a + (n - 1)d)

Substituting the value of a and d

a_n = 1/(3 + (n - 1)3)

= 1/(3 + 3n - 3)

= 1/(3n)

Finding the 5th term 

a_n = 1/(3n)

a₅ = 1/(3 × 5)

= 1/15

So, the 5th term of the sequence is 1/15

The use of explicit formula can be understood by the example mentioned below.

Find the 5th term of the sequence where a_n=3_n+2.

Step 1: Identify the formula and substitute n = 5n 

a₅ = 3(5)+2

Step 2: Simplify the expression.

a₅ = 15 + 2 = 17.

The fifth term of the sequence is 17.

Explicit formulas in math are a difficult topic to comprehend, therefore some tips and tricks are useful to master the topic.

Always Identify the Type of Sequence First: Before using any formula, check if it’s arithmetic, geometric, or harmonic.


Write Down the Given Data Clearly: Always list what is given, a_1, n, d, or r.


Substitute Carefully: When substituting into the formula, use parentheses to avoid sign or order errors.


Don’t Forget the Order of Operations: Follow BODMAS/PEDMAS rules while simplifying.


Watch for Negative or Fractional Values: If r<0 or d<0 o, terms may alternate signs.

To calculate or predict a specific term in a sequence, we use the explicit formulas. Here are some real-life applications of the explicit formulas.

• In sports, we use the arithmetic sequence to design a workout routine to plan a steady increase in the repetitions. At any point in training to determine the number of repetitions, we use the explicit formula. 

• In finance, to calculate the savings, loans, and investments, where the amount forms a geometric sequence, we can use the explicit formula to predict the amount at a particular time. 

• For linear population growth models, the explicit formula can predict the population at a specific time. 

• In computer algorithms, we can use explicit formulas to directly calculate the value in the sequence, which helps in improving efficiency. 

• In environmental studies, explicit formulas can be applied to model patterns such as temperature changes or rainfall distribution over time, helping predict future trends based on previous data.