![]() 2.20 Example 20įor a detailed solution, see here. 2.19 Example 19įor a detailed solution, see here. 2.18 Example 18įor a detailed solution, see here. 2.17 Example 17įor a detailed solution, see here. 2.16 Example 16įor a detailed solution, see here. 2.15 Example 15įor a detailed solution, see here. 2.14 Example 14įor a detailed solution, see here. 2.13 Example 13įor a detailed solution, see here. ![]() 2.12 Example 12įor a detailed solution, see here. 2.11 Example 11įor a detailed solution, see here. 2.10 Example 10įor a detailed solution, see here. 2.9 Example 9įor a detailed solution, see here. 2.8 Example 8įor a detailed solution, see here. 2.7 Example 7įor a detailed solution, see here. 2.6 Example 6įor a detailed solution, see here. 2.5 Example 5įor a detailed solution, see here. 2.4 Example 4įor a detailed solution, see here. 2.3 Example 3įor a detailed solution, see here. 2.2 Example 2įor a detailed solution, see here. Noting that can be expressed in terms of and as follows, Using the language of matrices from CP1 §6 Matrices, these may be written as Īnd is an arithmetic sequence with the common difference, i.e.Ĭomment 1: Having obtained a quadratic expression, we can demonstrate how a quadratic sequence produces a constant sequence for the second difference.Ĭomment 2: We can cross-check our result by solving the simultaneous equations, a linear function in ,Ĭonsider a sequence where we examine the differences and the second layer forms a constant sequence with the common difference. It is straightforward to recognise that this is a linear sequence, i.e. Where is the first term and is the common difference. The general term of an arithmetic sequence is given by Then, by using, we can determine the other coefficients.ġ.The coefficient is given by the constant divided by, i.e. If the -th layer is a constant sequence, the original sequence is a polynomial sequence of degree $m$ of the form In this section, we shall investigate where these rules come from and generalise the results further: Then, using and, we can find the values of and. In the example above, since the constant is 2, we find. If the second layer is a constant sequence, the original sequence is a quadratic sequence of the formĪlso, the value of is given by the constant divided by 2. If the first layer – the difference of the original sequence – is a constant sequence, the original sequence is an arithmetic/linear sequence of the form We notice that a constant sequence appears in the 2nd layer and, in GCSE, we postulated a set of rules: For example, consider the following sequence. ![]() In GCSE maths (Y10-11), we learnt linear/arithmetic and quadratic sequences. This post is a spin-off from CP1 §3 Series and CP2 §2.1 The method of differences. ![]()
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