Formulas To Solve Arithmetic Progression

Formula of Arithmetic Progression

nth term of an AP

t_n = a + (n – 1)d 
where t_n = 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

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

a = the first term,
d= common difference,
l = t_n = n^th 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 n^th is in linear expression then the sequence is in AP.
• If a₁, a₂, a₃, …, a_nand b₁, b₂, b₃, …, b_n, are in AP. then a₁+b₁, a₂+b₂, a₃+b₃, ……, a_n+b_n  and a₁–b₁, a₂–b₂, a₃–b₃, ……, a_n–b_n will also be in AP.
• If n^th term of a series is t_n = 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.