Series

Abstract

An ordered set of numbers forms a sequence, the order being determined by an algebraic formula. For example

• (a) 2, 5, 8, 11 is a sequence which could be formed from

  $$ {T_n} = 2 + 3\left( {n - 1} \right) = 3n - 1 $$

  T _n is the nth term. (This is an arithmetric progression.)

• (b) 2, 8, 11, 5 could be formed from ${T_n} = 3{n^3} - \frac{{45}}{2}{n^2} + \frac{{105}}{2}n - 31$.

• (c) 2, 10, 50, 250 could be formed from T _n = 2 × 5ⁿ⁻¹ (a geometric progression—G.P.).

• (d) 2, 1, ½, ¼ could be formed from T _n = 2 × (½)ⁿ⁻¹ = (½)ⁿ⁻² (another geometric progression).

• (e)
  $$\frac{1} {{1.3}},\frac{1} {{3.5}},\frac{1} {{5.7}},\frac{1} {{7.9}}\;is\;such\;that\;{T_n} = \frac{1} {{(2n - 1)(2n + 1)}}$$
• (f)
  $$1,\frac{1} {{2!}},\frac{1} {{3!}},\frac{1} {{4!}}\;is\;such\;that\;{T_n} = \frac{1} {{n!}}$$