Arithmetic Progression (A.P)

Arithmetic Progression(AP) is very important topic in Mathematics which can be used in programming to solve many problems.

An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. Difference here means the second term minus the first term.In simple terms, it means that the next number in the series is calculated by adding a fixed number to the previous number in the series.

For instance, the sequence 1,4,7,10,13,16..... is an arithmetic progression with common difference of 3.

( 16-13 = 13-10 = 10-7 = 7-4 = 4-1 = 3 )

Illustration of Arithmetic Progression. [Image Courtesy: GeeksforGeeks]

Terms related to Arithmetic Progression

Initial Term: The first term of the A.P is called the initial term. It is denoted by a_n. Common Difference: The value by which the consecutive terms increase or decrease is called the common difference. It is denoted by d.

NOTE: The behavior of the arithmetic progression depends on the common difference d. If the common difference is:

• positive, then the members (terms) will grow towards positive infinity.
• negative, then the members (terms) will grow towards negative infinity.

Formula of n^th term of the A.P

If a is the initial term and d is the common difference.Then, the explicit formula for the n^th term of the A.P is a_n = a + (n-1)d

Formula of the sum of n^th term of A.P

If a is the initial term and d is the common difference.Then, the explicit formula for the sum of n terms of the A.P is S_n = n/2 [2a + (n-1)d]

Properties of Arithmetic Progressions

Increasing/Decreasing Sequence:

• If the common difference is positive, i.e. d>0 then the arithmetic progression is an increasing sequence and satisfies the condition a_n-1 > a_n : a₁ < a₂ < a₃ < ... For example, the arithmetic progression with initial term 2 and common difference 3 i.e. 2,5,8,11,14,...., is an increasing sequence.
• If the common difference is negative, i.e. d<0 then the arithmetic progression is a decreasing sequence and satisfies the condition a_n-1 > a_n : a₁ > a₂ > a₃ > ... As an example, the arithmetic progression with initial term 1 and common difference -3 i.e.1,-2,-5,-11,..., is a decreasing sequence.

Other properties:

• If a,b,c are in AP, then 2b = a + c.
• If each term of an AP is increased, decreased, multiplied, or divided by a constant non-zero number, then the resulting sequence is also in AP.
• If the n^th term of any sequence is of the form an + b , then the sequence is in AP where the common difference is a.

Implementation

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