Arithmetic Progressions

An AP is a sequence where

An AP is a sequence where each term is obtained by adding a fixed number (common difference 'd') to the previous term. General form: a, a+d, a+2d, a+3

The fixed number added to each

The fixed number added to each term to get the next term. Found by: d = a(k+1) - ak. Can be positive, negative, or zero.

The general term of an AP

The general term of an AP is given by an = a + (n-1)d, where 'a' is first term, 'd' is common difference, and 'n' is the term number.

Sum of first n terms

Sum of first n terms: S = n/2[2a + (n-1)d] or S = n/2(a + l) where 'l' is the last term. Derived using Gauss's method.

If

If a, b, c are in AP, then b = (a+c)/2. The middle term 'b' is the arithmetic mean of 'a' and 'c'.