There are few arithmetic progression formulae used in solving the sequence in which the numbers are arranged in a definite order. In short the formulas used in arithmetic progression.

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**A sequence $a_{1}, a_{2},a_{3},a_{4}, …,a_{n}$ in in A.P.1) constant d = common difference = $a_{2} – a_{1}$2) If “a” is the first term and “d” is the common difference of an AP, then the A.P. isa, a + d, a + 2d, a + 3d, a + 4d , …3) The nth term $a_{n}$ of an A.P. with first term “a” and common difference “d” is given by$a_{n}$ = a + (n – 1) d4) Let there be an A.P. with first term “a” and common difference “d”. If there are “m” terms in the A.P. thennth term from the end = ( m – n + 1)th term from the beginning = a + ( m – n)dAlso, nth term from the end = last term + (n – 1) (-d) = l – (n – 1)d , where l = last term.5) Various terms is an A.P can be chosen in the following manner.(i) 3 terms —- a -d, a ,a + d —– common difference = d = d(ii) 4 terms —- a – 3d , a- d, a + d , a, a + 3d —- d = 2d(iii) 5 terms —- a – 2d , a- d, a, a + d , a + 2d —- d = d(iv) 6 terms —- a – 5d , a- 3d, a + d , a + 3d , a + 5d —- d = 2d6) When there are three numbers “a” , A and “b” are in A.P. then A is called arithmetic mean of the numbers “a” and “b”.A = $A = frac{a + b}{2}$**

## Arithmetic progression formulae on sum

7) The sum to “n” terms of an A.P. with first term “a” and common difference “d” is given by$S_{n}= frac{n}{2} <2a + (n - 1)d>$OR$S_{n}= frac{n}{2}$ where last term = l = a + ( n – 1)d8) Sum of first “n” natural numbers = $S_{n} = frac{n(n + 1)}{2}$9) Sum of squares of the first “n” natural numbers= $S_{n} = frac{n(n + 1)(2n + 1)}{6}$10) Sum of cubes of the first “n” natural numbers= $S_{n} = frac{

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