Properties of Arithmetic Progression

TheArithmetic Progression (A.P.) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference (d).

For Example:

The Sequence:

2, 5, 8, 11, 14 ....

It is an AP with the common difference (d) = 3.

Arithmetic Progression (A.P.) properties simplify calculations and provide a structured way to analyze sequences. From understanding how terms relate to common differences, to deriving sums and relationships, the properties of A.P. form the foundation for solving many mathematical problems.

Properties of Arithmetic Progression (AP) with Examples:

Adding/Subtracting a Constant

Adding or subtracting the same constant from all terms results in another A.P.

Example:

3, 6, 9, 12, 15, . . . (A.P)

Add 2 to each term gives: (+2)

5, 8, 11, 14, 17, . . . (A.P)

Similarly, this property can be applied for subtraction also it will give the AP.

Proof:

Let x₁, x₂, x₃, … be an Arithmetic Progression (A.P.) with a common difference d. Let a fixed constant k be added to or subtracted from each term of the A.P. The new sequence becomes:

x₁ + k, x₂ + k, x₃ + k . . . .
or
x₁ - k, x₂ - k, x₃ - k . . . .

Let y_n = x_n + k, where n =1, 2, 3, . . .. The new sequence is y₁, y₂, y₃, . . . ..

The difference between consecutive terms in the new sequence is:

y_{n + 1} - y_n = (x_n+1 + k) − (x_n + k)
y_{n + 1} - y_n = (x_n+1 - x_n) + (k- k)
y_{n + 1} - y_n = x_n+1 - x_n

Since x_n+1 - x_n = d for all n ∈ N:

y_n+1 - y_n = d

The new sequence y₁, y₂, y₃, is also an Arithmetic Progression, and the common difference d remains unchanged when a constant k is added to or subtracted from each term.

Multiplying/Dividing a Constant

Multiplying or dividing all terms of an A.P. by the same constant results in another Arithmetic Progression (A.P.), where a common difference is scaled.

Example:

3, 6, 9, 12, 15, . . . (A.P.)

Multiply each term by k = 2: (×2)

6, 12, 18, 24, 30, . . . (A.P.)

Here, the common difference will be multiplied by the same constant k × d

Proof:

Let x₁, x₂, x₃, . . . be an Arithmetic Progression (A.P.) with a common difference d. If each term of the A.P. is multiplied by a non-zero constant k, the new sequence becomes:

b₁ = x₁ ⋅ k, b₂ = x₂ ⋅ k, b₃ = x₃ ⋅ k, . . .

To Prove: The resulting sequence b₁, b₂, b₃, . . . is also an A.P., and its common difference is d′= k ⋅ d

The common difference of the new sequence is :

b_n+1​ − b_n ​= (x_n+1 ​⋅ k) − (x_n ​⋅ k)
b_n+1 ​− b_n​ = k ⋅ (x_n+1 ​− x_n​)

Since x_n+1 − x_n = d (the common difference of the original A.P.):

b_n+1 − b_n = k ⋅ d

Thus, the new sequence has a constant difference d′ = k ⋅ d

If each term of an A.P. is multiplied by a constant k, the resulting sequence is also an A.P., with the common difference scaled to d′ = k ⋅ d.

Note: If all terms of an AP are multiplied by a constant k, the resulting sequence is also an AP, with the common difference scaled to k ⋅ d. Similarly, if all terms are divided by k, the resulting sequence is an AP with a common difference of d/k.

The sum of Terms Equidistant from Beginning to End

In a finite Arithmetic Progression (A.P.), the sum of terms equidistant from the beginning and end is always equal to the sum of the first and last terms.

In simpler terms, if you pick two terms with the same distance from the beginning and the end of the sequence, their sum will always be the same. This sum is equal to the sum of the first term and the last term.

Example:

Consider the finite A.P.:

2, 5, 8, 11, 14, 17,

Here, a₁ = 2, a_n = 17, and n = 6

• Sum of the 1st and 6th terms:
  • a₁ + a₆ = 2 + 17 = 19
• Sum of the 2nd and 5th terms:
  • a₂ + a₅ = 5 + 14 = 19
• Sum of the 3rd and 4th terms:
  • a₃ + a₄ = 8 + 11 = 19

Combined Property for Middle Terms in an A.P.:

If the Number of Terms is Odd:

• The average of the first and last terms of an AP is equal to the value of the middle term:

If the number of Terms is Even:

• The average of the two middle terms is equal to the average of the first and last terms

Three Consecutive Terms in an A.P.:

Any three consecutive terms in an A.P. follow the property:

This means the middle term is the arithmetic mean of the terms surrounding it.

• For any Three terms x, y, and z are in an A.P., they satisfy:

2y = x + z