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# Series

Chapter
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## Abstract

An ordered set of numbers forms a sequence, the order being determined by an algebraic formula. For example
1. (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.)

2. (b)

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

3. (c)

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

4. (d)

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

5. (e)
$$\frac{1} {{1.3}},\frac{1} {{3.5}},\frac{1} {{5.7}},\frac{1} {{7.9}}\;is\;such\;that\;{T_n} = \frac{1} {{(2n - 1)(2n + 1)}}$$

6. (f)
$$1,\frac{1} {{2!}},\frac{1} {{3!}},\frac{1} {{4!}}\;is\;such\;that\;{T_n} = \frac{1} {{n!}}$$

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## Copyright information

© H. J. Halstead and D. A. Harris 1963

## Authors and Affiliations

1. 1.Mathematics DepartmentRoyal Melbourne Institute of TechnologyAustralia