Finite sequences and series

This chapter discusses finite sequences and series. A sequence or progression is a set of numbers in some definite order, the successive terms (or numbers) of the sequence being formed according to some rule. For the sequence of positive integers 1, 2, 3, 4,…, the r th term is the integer r ; for the sequence 1, 4, 9, 16, …, the r th term is the number r 2 . It is usual to denote the r th term of a general sequence by u r and the sequence by u 1 , u 2 , u 3 … u r . The rule defining a sequence is often given in the form of some formula for u r in terms of r , although this is not necessarily so. Thus, for first sequence, u r = r and for second sequence, u r = r 2 . On the other hand, a series is obtained by forming the sum of the terms of a sequence. A finite series is obtained if a finite number of terms of the sequence are summed. The sum of the first n terms of the sequence u 1 , u 2 , … is generally denoted by S n : S n = u 1 + u 2 + u 3 + … + u n . The chapter discusses the arithmetic and the finite geometric sequence and series. It also discusses the infinite geometric series.