Arithmetic Progression

An Arithmetic Progression (A.P.), commonly referred to as an arithmetic sequence, is a series of numbers where each subsequent term is derived by adding a fixed number to the preceding term. This consistent difference is termed the common difference of arithmetic progression. For example, a series of Integers: –3, –2, –1, 0, 1, 2, 3, . . . is an arithmetic progression whose common difference is 1.

In Mathematics, mainly there are three types of sequences, but they differ in their method of progression,

1. Arithmetic Progression
2. Geometric Progression
3. Harmonic Progression

1.0Arithmetic Progression Definition                                                                                                                            

An Arithmetic Progression (A.P.) is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term. This fixed constant difference is known as the common difference of  arithmetic progression.

The general form of an arithmetic progression is represented by the terms a, a + d, a + 2d, a + 3d, and so on. 

Where 'a' is the first term of the sequence and 'd' is the common difference. The terms of an AP form a pattern where each term is related to the previous one by a common difference.

2.0Common Difference of an Arithmetic Progression

The Common Difference of an A.P. is the fixed value added to each term to obtain the next term in the sequence. 

Note: The Common Difference of an AP can be positive, negative, or zero.

Example: Find the common Difference of an A.P –7, –5, –3, –1, . . .

Solution: = $\mathrm{d}={\mathrm{a}}_2-{\mathrm{a}}_1$

d=-5-(-7)=-5+7=2

3.0General Form of an A. P

The general form of an arithmetic progression (AP) is a representation that allows us to express any term in the progression without explicitly listing all the terms. It is a concise and systematic way to describe the sequence based on its first term ('a') and the common difference ('d').

The general form of an arithmetic progression can be written as:

a, a + d, a + 2d, a + 3d, . . . 

Where 'a' is the first term of the sequence and 'd' is the common difference. 

4.0N^th term of an Arithmetic Progression

Using this general form, we can find any term in the arithmetic progression by knowing its position or index in the sequence. The nth term of the AP (denoted by 'a_n') can be expressed as:

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

This formula is also called the n^th term of an A. P

5.0Sum of an Arithmetic Progression

The Sum of the First 'n' terms within an Arithmetic Progression (AP), denoted by S_n, can be calculated using the sum formula for AP. The formula used to calculate the sum of 'n' terms in the sequence is derived based on the general form of an AP and is given by:

${\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}[2\mathrm{a}+(\mathrm{n}-1)\mathrm{d}]$

Here:

• 'S_n' denotes the cumulative sum of the initial 'n' terms in the Arithmetic Progression (AP).
• 'n' is the number of terms whose sum we want to find.
• 'a' is the first term of the AP.
• 'd' represents the common difference of AP

This formula is derived using the concept that the sum of an AP is essentially the average of the first term and last term multiplied by the number of terms. By substituting the values of 'a', 'd', and 'n' into the formula, we can easily find the sum of any number of terms in the given arithmetic progression.

Sum of an Arithmetic Progression when the Last Term is Given

If in an A.P, the First Term, Common Difference, and Last Term are given, then the Sum of an A. P is given as:

S_n = (n/2) (first term + last term)

6.0Arithmetic Progression Formulas

7.0Difference Between Arithmetic Progression and Geometric Progression

A Geometric Progression (G.P.), commonly referred to as a geometric sequence, is a series of numbers in which each term, except the first, is obtained by multiplying the preceding term by a constant non-zero value known as the common ratio (represented by 'r'). The general form of a geometric progression is given by:

a, ar, ar², ar³, . . .

Here,

• 'a' is the first term of the geometric progression,
• 'r' is the common ratio.

The nth term of a geometric progression (denoted by T_n) is given by

T_n = a⋅r⁽ⁿ⁻¹⁾ 

The cumulative sum of the initial 'n' terms in a geometric progression (denoted by S_n) can be found using the formula:

S_n = a(rⁿ−1)/r−1

Difference Between Arithmetic and Geometric Progression is:

8.0Arithmetic Progression Solved Examples

Examples 1: Find the First Term and Common Difference of the AP: 2, 5, 8, 11, 14, . . .

Solution: First Term (a):  2

Common Difference (d): 5 – 2 = 3

Example 2: Find the tenth term of the AP: 2, 7, 12, . . .

Solution: We know that, a_n = a + (n – 1) × d

a = 2; n = 10; d = 7 – 2 = 5

a₁₀ = 2 + (10 – 1) × 5

a₁₀ = 2 + (9) × 5

a₁₀ = 2 + 45

a₁₀ = 47

Example 3: Determine the AP whose fourth term is 7 and the eighth term is 15.

Solution: We know that a_n = a + (n – 1) × d

a₃ = a + (4 – 1) d = a + 3d = 7 . . . (1) 

and a₇ = a + (8 – 1) d = a + 7d = 15 . . . (2)

Solving Eq 1 and Eq 2 

We get a = 1, d = 2.

Hence, the required AP is 1, 3, 5, 7, 9, 11, 13, . . . 

Example 4: How many terms of the AP: 24, 20, 16,12, . . . must be taken so that their sum is 80?

Solution: Here, a = 24, d = 20 – 24 = –4, S_n = 80. 

We need to find n. 

We know that S_n = (n/2)[2a + (n − 1) × d]

So, 80 = (n/2)[48 + (n − 1) ( –4)] 

80 = (n/2) [52 – 4n]

or 80 = 26n – 2n²

or 40 = 13n – n²

or n² – 13n + 40 = 0  

or (n – 5) (n – 8) = 0 

or n = 5 or 8 

Both values of n are admissible. So, the number of terms is either 5 or 8.

Example 5: Which term of the AP: 121, 117, 113, . . ., is its first negative term?

Solution: Here, a = 121, d = 117 – 121 = –4,  

a_n = a + (n – 1) × d

      = 121 + (n – 1) ( –4)

      = 121 – 4n + 4

      = 125 –4n

We need to find first negative term in this AP

Therefore, nth term will be less than zero.

 a_n < 0

125 – 4n < 0

125 < 4n

n > 125/4

n > 31.25

Therefore, the 32^nd term will be the first negative term of this AP.

Example 6: How do you find the sum of the first 'n' terms of an arithmetic progression?

Solution: The sum of the first 'n' terms of an arithmetic progression can be calculated using the formula-

${\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}[2\mathrm{a}+(\mathrm{n}-1)\mathrm{d}]$

where:

• S_n is the sum of 'n' terms in AP,
• 'n' is the number of terms,
• 'a' is the first term,
• 'd' is the common difference.

9.0Arithmetic Progression Practice Problems  

Question 1: Find the First Term and Common Difference of the AP: 12, 15, 18, 21, 24, . . .

Question 2: Find the 20th term of the AP: –3, 1, 5, . . .

Question 3: Determine the AP whose 2nd term is 7 and the 16^th term is 35.

Question 4: How many terms of the AP: 9, 17, 25, . . . must be taken to give a sum of 636?