Sequences As Functions, Example 3

GeoGebra Commands & Parameters

1. LaTeX(steps) → counter
2. LaTeX("\text{Move the slider to solve the problem.}") → text0
3. LaTeX("\text{Use the buttons below to return to the original}\cr\text{problem or randomly generate a similar practice}\cr\text{problem.}") → RandomizationText
4. Corner(2, 3) → topright
5. Corner(2, 1) → bottomleft
6. If(const > 0, "+", "-") → constSign
7. If(choose ≟ 0, LaTeX[(((("a_n=" * coef) * "n") * constSign) * absConst)], LaTeX[(((((("a_n=" * coef) * "a_{n-1}") * constSign) * absConst) * "\text{ and }a_0=") * const)]) → formula
8. If(choose ≟ 0, explicit, recursive) → sequenceVals
9. Sequence((n(j), sequenceVals(j)), j, 1, graphterms) → plotPoints
10. LaTeX("\text{Determine whether this is an explicit formula or a}\cr\text{recursive formula.}") → text1
11. LaTeX((((((("\text{Find and graph the first " * graphterms) * " terms, starting with }\textcolor{#000099}{n=1}\text{,}\cr\text{of the sequence given by the formula }\cr\textcolor{#000099}{") * formula) * "}\text{,}\cr\text{where }\textcolor{#000099}{a_n}\text{ is the value of the term and }\textcolor{#000099}{n}\text{ is the term}\cr\text{number. Then find the ") * finalterm) * "th term.}")) → Problem
12. If(choose ≟ 0, LaTeX["\text{This is an explicit formula because the right hand}\cr\text{side depends only on }\textcolor{#000099}{n}\text{ and not on }\textcolor{#000099}{a_{n-1}}\text{ or similar}\cr\text{terms.}"], LaTeX["\text{This is a recursive formula because the right hand}\cr\text{side depends on }\textcolor{#000099}{a_{n-1}}\text{ and we are given an initial}\cr\text{term, }\textcolor{#000099}{a_0}\text{.}"]) → text2
13. LaTeX((("\text{Find the first " * graphterms) * " terms of the sequence.}")) → text3
14. If(choose ≟ 0, Sequence[LaTeX[(((((((((("a_" * k) * "=") * coef) * "(") * k) * ")") * constSign) * absConst) * "=") * sequenceVals(k))], k, 1, graphterms], Append[LaTeX[(((((((("a_1=" * coef) * "(") * const) * ")") * constSign) * absConst) * "=") * sequenceVals(1))], Sequence[LaTeX[(((((((((("a_" * k) * "=") * coef) * "(") * sequenceVals(k - 1)) * ")") * constSign) * absConst) * "=") * sequenceVals(k))], k, 2, graphterms]]) → firstTerms
15. LaTeX((("\text{Substitute the respective values for }\textcolor{#000099}{n=1}\text{ through }\cr\textcolor{#000099}{n=" * graphterms) * "}\text{.}")) → text4
16. LaTeX(formula) → text4b
17. TableText({firstTerms}, "v") → text4c
18. Sequence(sequenceVals(k), k, 1, graphterms - 1) → shortList
19. LaTeX("\text{Create ordered pairs from the sequence.}") → text5
20. LaTeX("\text{The first coordinate is }\textcolor{#000099}{n}\text{ and the second coordinate}\cr\text{is }\textcolor{#000099}{a_n}\text{.}") → text6
21. LaTeX("\text{Plot the ordered pairs.}") → text8
22. LaTeX((("\text{Find the " * finalterm) * "th term in the sequence.}")) → text9
23. If(choose ≟ 0, LaTeX[(((("\text{Since the sequence is explicitly defined, evaluate}\cr\text{the formula, }\textcolor{#000099}{" * formula) * "}\text{ at }\textcolor{#000099}{n=") * finalterm) * "}\text{.}")], LaTeX[(("\text{Since the sequence is recursively defined, continue}\cr\text{building up to the " * finalterm) * "th term one term at a time.}")]) → text10a
24. Sequence(sequenceVals(k), k, 1, graphterms) → graphedList
25. Max(0, Max[graphedList]) → maxy
26. Sequence(LaTeX[(((((((((("a_" * k) * "=") * coef) * "(") * sequenceVals(k - 1)) * ")") * constSign) * absConst) * "=") * sequenceVals(k))], k, graphterms + 1, finalterm) → moreTerms
27. If(choose ≟ 0, LaTeX[(((((((((("a_" * finalterm) * "=") * coef) * "(") * finalterm) * ")") * constSign) * const) * "=") * sequenceVals(finalterm))], TableText[{moreTerms}, "v"]) → text10b
28. LaTeX((((("\text{The " * finalterm) * "th term in the sequence is ") * sequenceVals(finalterm)) * ".}")) → text10c
29. LaTeX((((((("\text{" * text10a) * "}\cr\textcolor{#000099}{") * text10b) * "}\cr\text{") * text10c) * "}")) → text10
30. LaTeX((((((("\text{The first " * graphterms) * " values are ") * shortList) * "}\cr\text{and ") * sequenceVals(graphterms)) * ".}")) → text4d
31. TableText({Append[LaTeX["(n,a_n)"], plotPoints]}, "v") → text7
32. Min(0, Min[graphedList]) → miny