- An arithmetic progression is a sequence of numbers in which each term is derived from the preceding term by adding or subtracting a fixed number called the common difference "d"

For example, the sequence 9, 6, 3, 0,-3, .... is an arithmetic progression with -3 as the common difference. The progression -3, 0, 3, 6, 9 is an Arithmetic Progression (AP) with 3 as the common difference. - The general form of an Arithmetic Progression is a, a + d, a + 2d, a + 3d and so on. Thus nth term of an AP series is T
_{n}= a + (n - 1) d, where T_{n}= n^{th}term and a = first term. Here d = common difference = T_{n}- T_{n-1}. - Sum of first n terms of an AP: S =(n/2)[2a + (n- 1)d]
- The sum of n terms is also equal to the formulawhere l is the last term.
- T
_{n}= S_{n}- S_{n-1}, where T_{n}= n^{th}term - When three quantities are in AP, the middle one is called as the arithmetic mean of the other two. If a, b and c are three terms in AP then b = (a+c)/2

- A geometric progression is a sequence in which each term is derived by multiplying or dividing the preceding term by a fixed number called the common ratio. For example, the sequence 4, -2, 1, - 1/2,.... is a Geometric Progression (GP) for which - 1/2 is the common ratio.
- The general form of a GP is a, ar, ar
^{2}, ar^{3}and so on. - The nth term of a GP series is T
_{n}= ar^{n-1}, where a = first term and r = common ratio = T_{n}/T_{n-1}) . - The formula applied to calculate sum of first n terms of a GP:
- When three quantities are in GP, the middle one is called as the geometric mean of the other two. If a, b and c are three quantities in GP and b is the geometric mean of a and c i.e. b =√ac
- The sum of infinite terms of a GP series S
_{∞}= a/(1-r) where 0< r<1. - If a is the first term, r is the common ratio of a finite G.P. consisting of m terms, then the nth term from the end will be = ar
^{m-n}. - The nth term from the end of the G.P. with the last term l and common ratio r is l/(r
^{(n-1)}) .

- A series of terms is known as a HP series when their reciprocals are in arithmetic progression.

Example: 1/a, 1/(a+d), 1/(a+2d), and so on are in HP because a, a + d, a + 2d are in AP. - The n
^{th}term of a HP series is T_{n}=1/ [a + (n -1) d]. - In order to solve a problem on Harmonic Progression, one should make the corresponding AP series and then solve the problem.
- nth term of H.P. = 1/(nth term of corresponding A.P.)
- If three terms a, b, c are in HP, then b =2ac/(a+c).

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- Sum of first n natural numbers =
- Sum of squares of first n natural numbers =
- Sum of cubes of first n natural numbers =