Algebra recursive sequence12/6/2023 ![]() The nature of $p(x^2)q(x^2)$ is to have no odd terms, so the corresponding recurrence is has even terms depend only on previous even terms, and odd terms only on previous odd terms. Your case is the particularly easy case, $p(x)=x-2, q(x)=x-3.$ Using the same geometric sequence above, find the sum of the geometric sequence through the 3 rd term. The equation for calculating the sum of a geometric sequence: a × (1 - r n) 1 - r. (Sometimes you can find a smaller common recurrence, but this one always works.) Comparing the value found using the equation to the geometric sequence above confirms that they match. More generally, if $x_i$ is a recurrence corresponding to $p(x)$ and $y_i$ aecurrence correspondind to $q(x)$ then $$(x_1,y_1,x_2,y_2,\dots)$$ In this case, the differences between consecutive terms are. Good question Well, the key pieces of information in both the explicit and recursive formulas are the first term of the sequence and the constant amount that you change the terms by, aka the common ratio (notice: the name 'common ratio' is specific to geometric sequences, the name that applies to arithmetic seq. Plug your numbers into the formula where x is the slope and you'll get the same result: 5 + x (10 1) 59. To find the recursive formula, start by looking at the differences and ratios of consecutive terms. ![]() Each number in the sequence is called a term (or sometimes 'element' or 'member'), read Sequences and Series for more details. Your shortcut is derived from the explicit formula for the arithmetic sequence like 5 + 2 (n 1) a (n). ![]() This answer assumes you know how the closed formula for $x_n$ corresponds to the roots of the polynomial: A Sequence is a set of things (usually numbers) that are in order.
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