Formulas To Solve Arithmetic Progression

Formulas to Solve Arithmetic Progression problems in Aptitude Test

Definition & Formulas for Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers such that the common difference between any two succeeding numbers is a constant. This constant is usually denoted by ‘d’ and is called ‘common difference’. The common difference ‘d’ can be zero, positive, and negative. Each number in the sequence is known as ‘term’ and the first term of the sequence is called ‘first term’ which is denoted by ‘a’.
The generally form of arithmetic progression(AP) a, a+d, a+2d, a+3d……. so on. So, the nth term of an AP series is Tn = a+ (n-1)d, where Tn = nth term and, ‘a’= first term and ‘d’= common difference.

formula for arithmetic progression
General form of an APa, a + d, a + 2d, a + 3d ………
First Terma
Common Differenced

For example:

Given sequence or series is 2, 4, 6, 8, 10,…

Here, a = 2 and d = 2

Finite or Infinite Arithmetic Progressions

Finite Arithmetic Progression

When there are limited number of terms in the sequence then it is known as Finite Arithmetic Progression.

For example:

10, 20, 30, 40, 50

Infinite Arithmetic Progression

When there are unlimited number of terms in the sequence then it is known as Infinite Arithmetic Progression.

For example:

3, 5, 7, 9, 11, 13, ..…….

Formula of Arithmetic Progression

nth term of an AP

tn = a + (n – 1)d 
where tn = nth term, a= the first term , d= common difference, n = number of terms in the sequence.

Number of terms in an AP

n= [(l-a)/d] + 1
where n = number of terms, a= the first term, l = last term, d= common difference.

Sum of first n terms in an AP

Sn = n/2 [2a + (n − 1) d] OR n/2  (a+l)
where,

a = the first term,
d= common difference,
l = tn = nth term = a + (n-1)d

Arithmetic Mean

If a, b, c are in AP, then there Arithmetic mean is

b = 1/2 (a + c)

Some other important formulas of Arithmetic Progression

Sum of first n natural numbers

S = n (n+1)/2

Sum of squares of first n natural numbers

S= {n(n+1)(2n+1)}/6

Sum of first n odd numbers

S =n2

Sum of first n even numbers

S = n(n+1)

Properties of Arithmetic Progression

  • If a fixed number is added or subtracted from each term of an AP, then the resulting sequence is also an AP and it has the same common difference as that of the original AP.
  • If each term in an AP is divided or multiply with a constant non-zero number, then the resulting sequence is also in an AP.
  • If nth is in linear expression then the sequence is in AP.
  • If a1, a2, a3, …, anand b1, b2, b3, …, bn, are in AP. then a1+b1, a2+b2, a3+b3, ……, an+band a1–b1, a2–b2, a3–b3, ……, an–bn will also be in AP.
  • If nth term of a series is tn = An + B, then the series is in AP 
  • Three terms of the A.P whose sum or product is given should be assumed as a-d, a, a+d.
  • Four terms of the A.P. whose sum or product is given should be assumed as a-3d, a-d, a+d, a+3d.

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