The sum S_{n }of the first n terms of an A.P. is given by

\( S_n= \frac{n}{2}(2a + (n – 1) d) \)

The arithmetic mean A of any two numbers a and b is given by \( \frac{a + b}{2}\) i.e., the sequence a, A, b is in A.P.

A sequence is said to be a geometric progression or G.P., if the ratio of any term to its preceding term is same throughout. This constant factor is called the common ratio. Usually, we denote the first term of a G.P. by a and its common ratio by r. The general or the nth term of G.P. is given by a_{n}= ar^{n – 1} The sum S_{n} of the first n terms of G.P. is given by \( S_n= \frac{a(r^n – 1)}{(r – 1)} = \frac{a(1 – r^n)}{(1 – r)}, {\rm if } r \ne 1 \).

The geometric mean (G.M.) of any two positive numbers a and b is given by \( \sqrt{ab}\) i.e., the sequence a, G, b is G.P.