Arithmetic Progression

An arithmetic progression (AP) is a sequence in which each term is obtained by adding a constant to the previous term. In other words, the difference between two consecutive terms in an AP will be the same. The difference is known as the common difference (d), and the first term is denoted by a or \(a_1\), and the n^th term is denoted by \(a_n\).

For example, 3, 6, 9, 12, 15, … is an arithmetic progression. Here, 'a' is 3, and the constant 'd' is also 3. In its general form, AP can be expressed as \(a, a + d, a + 2d,…, a + (n - 1)d\).

Common Terms in an Arithmetic Progression (AP)
 

The commonly used terms in arithmetic progression are: 

First term (a or \(a_1\)): 

It is the first value in an arithmetic progression. It determines where the sequence begins. For example, in the AP: 5, 8, 11, 14. . . the first term a = 5. 

Common difference(d): 

The common difference is the constant value added to each term to get the next term. It is calculated by subtracting any term from the term that follows it. 

\(D = a_2 - a_1\).

For example, in the AP: 10, 7, 4, 1, …. the common difference (d) is -3, showing that the sequence is decreasing. 
 

nth term (\(a_n\)): 

The nth term represents any specific term in the AP and can be used to find a term at a particular position without writing all the previous terms. It is given by the formula:
\(a_n = a + (n - 1)d\)
 

Key formulas for arithmetic progression for calculating common difference, the nth term, and the sum of terms. 

 

The Common Difference in AP

The common difference of a sequence is the difference between two consecutive terms, and it is denoted by  'd':

\( d = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = \dots = a_n - a_{n-1} \)
 

The nth Term of an AP

The nth term of an AP can be expressed as \( a_n = a + (n - 1)d \)

Where n is the number of terms

a is the first term 

\(a_2\) is the second term

\(a_3\) is the third term

d is the common difference

\(a_n\) is the nth term of the sequence
 

Sum of the first n Terms of an AP
 

Now we will learn how to find the sum of the first n terms of an AP. So let’s consider the first n terms of an AP as:

\( a_1, \; a_1 + d, \; a_1 + 2d, \; a_1 + 3d, \; \dots, \; a_1 + (n-1)d \), and the sum of these n terms is \(s_n\)

Then the sum of nth terms:

\( S_n = a_1 + (a_1 + d) + (a_1 + 2d) + (a_1 + 3d) + \dots + \big(a_1 + (n-1)d\big) \)

Now let's reverse the sequence and add corresponding terms. It can be written as:

\( S_n = a_n + (a_n - d) + (a_n - 2d) + (a_n - 3d) + \dots + \big(a_n - (n-1)d\big)\)

Adding these two equations: 

\( S_n = a_1 + (a_1 + d) + (a_1 + 2d) + (a_1 + 3d) + \dots + \big(a_1 + (n-1)d\big) \)

\( S_n = a_n + (a_n - d) + (a_n - 2d) + (a_n - 3d) + \dots + \big(a_n - (n-1)d\big) \)

By adding these, we get: \( 2S_n = (a_1 + a_n) + (a_1 + a_n) + (a_1 + a_n) + \dots + (a_1 + a_n) \)

All the d terms are cancelled out here, so:

\( 2S_n = n(a_1 + a_n) \)

\( S_n = \frac{n(a_1 + a_n)}{2} \), let's consider it as equation 1

Substituting the formula to find an in equation 1, \( a_n = a_1 + (n - 1)d \)

Then, \( S_n = \frac{n}{2} \Big(a_1 + a_1 + (n - 1)d\Big) \) 

\( S_n = \frac{n}{2} \Big[ 2a + (n - 1)d \Big] \), let’s consider this equation as 2

Equations 1 and 2 are used to find the sum of an arithmetic progression.

Hence, proved the arithmetic progression sum proof. 

So, Sum of nth term: \( S_n = \frac{n}{2} \Big[ 2a + (n - 1)d \Big] \) and \( S_n = \frac{n}{2} (a + l) \)

The nth term of an AP is used to find any term in the sequence without listing all the previous terms. There is a formula that we can use to find the nth term, which is \(a_n = a + (n - 1)d \). 

For example, for the AP: 2, 4, 6, 8, 10,… we find the nth term.

Here, the first term a = 2 and the common difference d = 2.

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

\(= 2 + (n - 1)2\)

\(= 2 + 2n -2\)

\(a_n = 2n\)

Use of AP Formula for General Term

To find the nth term of an AP, we use the formula: \(a_n = a + (n - 1)d\)

For example, for the sequence 10, 20, 30, 40, …, here, a = 10 and d = 10. 

So, the 5th term, \(a_5 = 10 + (5 - 1)10 = 10 + 40 = 50\)

Similarly, to find the 55th term, we can use the formula,
 

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

\(a_{55} = 10 + (55 - 1) \times 10 \)

\(= 10 + 54 × 10 \)

\(= 10 + 540 = 550. \)

So, the 55th term of this AP is 550.
 

The arithmetic sequence explicit formula is the nth term of an AP, it is used to find any term of the sequence.

Now we will learn to find the sum of an arithmetic progression. The sum of the first ‘n’ terms can be calculated with the help of the formula mentioned below.

When the nth term is unknown, the sum of n is calculated using the formula:

\(S_n = \frac{n}{2} \left( 2a + (n - 1)d \right) \)

When the nth term is known, the sum of n is calculated using: \(S_n = \frac{n}{2} (a + a_n) \)

Where 'n' is the number of terms,

'a' is the first term,

'd' is the common difference, and

\(a_n\) is the nth term.

Let us now understand how to use the formulas for the sum of n terms of an AP through the following examples. 
 

Example 1: Find the sum of the first 10 terms of the AP: 2, 5, 8, 11, …

Here, first term, a = 2
Common difference, d = 3
Number of terms, n = 10.

Using the formula: 


 \(\begin{align} S_n &= \frac{n}{2}\,[2a + (n - 1)d] \\ S_{10} &= \frac{10}{2}\,[2(2) + (10 - 1)3] \\ S_{10} &= 5\,[4 + 27] \\ S_{10} &= 5 \times 31 \\ S_{10} &= 155 \end{align} \)

So, the sum of the first 10 terms is 155. 
 

Example 2: Find the sum of the first 15 terms of an AP where the first term, a = 7, and the 15th term, \(a_{15} = 49\)

Using the formula: 
 

\(\begin{align}\ S_n &= \frac{n}{2}(a + an)\\ \\S_{15} &= \frac{15}{2} × 56\ \\&= 15 × 28 = 420\end{align}\)

There are mainly two types of progressions: arithmetic and geometric progressions (GP). Here, we will be discussing the difference between AP and GP:

Here are some tips and tricks for the children and their parents to master arithmetic progression. 
 

• Start by understanding simple concepts. Arithmetic progression is a sequence of numbers where the difference between one number and the next is always the same. 
   
• Parents must try to use real-life examples like counting steps, money saving, or chocolates to make it relatable and interesting for their child to remember well. 
   
• Visual learning can be helpful in many ways. Try to draw a number line to show numbers are points and highlight the equal jumps. Use objects like coins, blocks, or breads to demonstrate sequences physically.
   
• Parents should as their children to find the difference between consecutive numbers. Ask them to practice with increasing and decreasing sequences.
   
• Use fun word problems that are interesting for kids. Make the learning process interactive by using real objects and letting them do the counting.
  
   
• Parents and teachers can encourage students to identify the first term (a) and the common difference (d) before solving any AP problem. This will help reduce mistakes in longer questions and provide greater clarity. 

• Make use of tables or charts to write the arithmetic progression terms in order. This helps students to visually observe the patterns and understand how each term increases or decreases by the same amount. 

• Ask students to predict the following terms in an arithmetic progression before finding them using formulas. This will strengthen their logical thinking and understanding. 

• Parents and teachers can connect AP to daily routines, such as weekly increases in pocket money or daily step counts, to show how sequences naturally occur in real life.

• Introduce increasing and decreasing APs separately to students. This will help students to compare both types and will clearly understand the role of positive and negative common differences.

In our real world, we use arithmetic progression from basic counting to calculate the interest rate. Here are some real-life applications of arithmetic progression:
 

• We use arithmetic progression to predict the upcoming term in a sequence. This is achieved by adding a constant difference. For e.g., we can predict the next number in the sequence 3, 7, 11,.....
   

• Exercise routines can be calculated using arithmetic progression. For example, increasing the number of push-ups by 2 each day: 2, 4, 6, 8… Distance or speed while cycling or running gradually increasing every week can follow an arithmetically progressive pattern.

• Heights of steps or number of bricks in successive rows may follow an arithmetic progression. Similarly, rows in an auditorium or theater often increase by a fixed number to maintain visibility.
   
• For seating arrangements, such as arranging seats in rows and columns with equal distance, we use arithmetic progression.
   
• In schools and colleges, increasing study hours by a fixed number of minutes each day: 30 min, 40 min, 50 min… Also, the marks or points awarded in some games or quizzes can follow arithmetic progression.
   
• We can watch over some plants that grow a fixed number of centimeters each week. The number of animals in successive generations sometimes grows linearly, forming arithmetically progressive patterns.