Arithmetic Sequence

An arithmetic sequence is a list of numbers where the difference between any two successive terms is the same. This constant difference is called common difference.

For example, in the arithmetic sequence given below, every term is obtained by adding 4 to its previous term.

4, 8, 16, 20, . . . 

Here, the common difference, denoted by (d), is 4.
 

To continue an arithmetic sequence, follow the given steps.

Step 1: Look at two consecutive terms. Pick any pair neighboring each other.

Step 2: Find the common difference (d).

Step 3: Check whether the sequence is rising \((d > 0)\), steady \((d = 0)\) or falling \((d < 0)\).

Step 4: Find the next term by adding the common difference to the last term.

Step 5: Repeat the addition to continue the sequence.
 

• To find a term at any position in the sequence, use the n^th term formula.

The formula for an arithmetic sequence is as follows:

\(a_n = a_1 + (n - 1) \times d \)

Here,
 

• a_n is the general or nth term
   
• a₁ stands for the first term
   
• n is the position of the term, and 
   
• d is the common difference.

To understand the formula better, let’s take an example:
 

2, 8, 14, 20, 26, ....

In the above sequence, d is 6. 

• \(a_1 = 2 \)
   
• \(a_2 = 2 + 6 \)
   
• \(a_3 = 2 + (2 × 6) \)
   
• \(a_4 = 2 + (3 × 6) \), and so on.
  .
  .
  .
  .
• \(an = a_1 + (n - 1) × d \)

The value at a specific position in an arithmetic sequence is represented by the nth term. The following formula can be used to find it:

          \(   a_n = a_1 + (n - 1) × d     \)   

Where, 
 

• a_n = nth term, 
   
• a₁ = first term, 
   
• and 𝑑 is the common difference between the terms.

For example, a sequence like 5, 9, 13, 17,..... each number rises by 4. Hence, the first term a1 is 5, and the common difference (d) is 4. Let’s substitute the equation to determine the seventh term, a₇:

\(a_7 = 5 + (7 - 1) × 4 \)

     \(  = 5 + (6 × 4) \)

     \( = 5 + 24\)

     \(= 29 \)

An arithmetic sequence's recursive formula is written as:

\(a_n = a_{n-1} + d\)

Where,
 

• a_n is the general or nth term,
   
• a_n-1 is the preceding term;
   
• 𝑑 is the common difference between terms.
   

The initial term (a₁) must be utilized to apply the recursive formula. In the sequence 2, 5, 8, 11, and so on, for instance, the first term is 2, and the common difference is 3.
 

The recursive formula then is \(a_1 = 2 \) and \(a_n = a_{n-1} + 3\) for n > 1. It means that to calculate a new term, we have to add 3 to the previous term.

The sum of an arithmetic sequence is resulted by adding all the terms of the sequence. The formula to calculate the sum of an arithmetic sequence is given below:
 

                           \(S_n = \frac{n}{2} \times (a + l) \)

Where,
 

• S_n is the sum of the sequence up to the nth term
   
• a is the first term
   
• l is the last term, and 
   
• n is the number of terms.

Alternatively, we can also use the below-mentioned formula if we know the first term 𝑎, the common difference 𝑑, and the number of terms 𝑛:

                                                
    \(S_n = \frac{n}{2} \times [2a + (n - 1)d] \)

Where,
 

• S_n is the sum of all the terms
   
• a is the first term,
   
• d is the common difference, and 
   
• n is the number of terms.
   
• 2a is 2 multiplied by the first term (when a is the first term of the sequence)

Understanding arithmetic sequences is easier when you know the right techniques. Use these helpful tricks to find common differences, calculate terms, and confidently tackle sequence-based math problems with ease.

• Always identify the pattern first, as every term changes by the same difference.
   
• Use subtraction to find the common difference quickly.
   
• Use the nth term formula to reach to any term instead of listing them out one by one.
   
• To find the total sum, use \(S_n = \frac{n}{2} (2a + ( n - 1) d) \). This is a shortcut to find the total quickly.
   
• Check d for negative and positive signs to identify falling and rising sequences, respectively.

Arithmetic sequences show up in a lot of real-life scenarios. Knowing how they work will help us make accurate decisions in a structured and mathematical way. 

• Monthly Savings and Budgeting
  
  When an individual sets aside $2000 every month, the savings form an arithmetic sequence: $2000, $4000, $6000, and so on. This pattern aids in financial planning by enabling individuals to forecast their savings after a designated number of months.

• Building and designing stairs
  
  Structures with evenly increasing levels, like stairs, often follow a pattern. E.g., each step on a ladder might be 6 inches higher than the one before it. Architects and builders can estimate the total rise, the number of steps, and the materials required more easily with this steady rise. 

• Plans for mobile data or subscriptions
  
  Some mobile plans or subscription services offer perks that keep increasing over time. As an example, a person might get 1GB of internet data in the first month, 2GB in the second, 3GB in the third, and so on. Users can then choose the right plan based on their knowledge about future data limits or service benefits.

• Tracking student performance
  
  Suppose a student improves their score on each test by the same number of points, say 5 points each time. This is called an arithmetic sequence. For instance, their scores could be 60, 65, 70, 75, and so on. 

• Seating arrangements in theaters and halls
  
  Many theaters and stadiums are built in such a way that each row has more seats than the one before it. For example, each row might have two more seats than the row before it. This forms a pattern and an arithmetic sequence, making it easier for engineers to construct these halls or stadiums.