Abstract
An ordered set of numbers forms a sequence, the order being determined by an algebraic formula. For example
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(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.)
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(b)
2, 8, 11, 5 could be formed from \( {T_n} = 3{n^3} - \frac{{45}}{2}{n^2} + \frac{{105}}{2}n - 31 \).
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(c)
2, 10, 50, 250 could be formed from T n = 2 × 5n−1 (a geometric progression—G.P.).
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(d)
2, 1, ½, ¼ could be formed from T n = 2 × (½)n−1 = (½)n−2 (another geometric progression).
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(e)
$$\frac{1} {{1.3}},\frac{1} {{3.5}},\frac{1} {{5.7}},\frac{1} {{7.9}}\;is\;such\;that\;{T_n} = \frac{1} {{(2n - 1)(2n + 1)}}$$
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(f)
$$1,\frac{1} {{2!}},\frac{1} {{3!}},\frac{1} {{4!}}\;is\;such\;that\;{T_n} = \frac{1} {{n!}}$$
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© 1963 H. J. Halstead and D. A. Harris
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Halstead, H.J., Harris, D.A. (1963). Series. In: A Course in Pure and Applied Mathematics. Palgrave, London. https://doi.org/10.1007/978-1-349-81604-0_2
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DOI: https://doi.org/10.1007/978-1-349-81604-0_2
Publisher Name: Palgrave, London
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