Arithmetic Progression formulae

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.
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. is
a, 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) d

4) Let there be an A.P. with first term 'a' and common difference 'd'. If there are 'm' terms in the A.P. then
nth term from the end = ( m - n + 1)th term from the beginning
                 = a + ( m - n)d
Also, 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 = 2d

6) 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
$S_{n}= \frac{n}{2}$ [a + l] where last term = l = a + ( n - 1)d

8) 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
= $S_{n} = \frac{[n(n + 1)]^{2}}{4}$


11th grade math

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