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That argument will \ be called ", StyleBox["x", FontSlant->"Italic"], " in the body of the function, which gives a result by adding one" }], "Text", CellChangeTimes->{{3.763744615304699*^9, 3.763744640707953*^9}},ExpressionUUID->"a7959692-9821-48c3-8870-\ fd69085b7717"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"addOne", "[", "2", "]"}]], "Input", CellChangeTimes->{{3.763744642023451*^9, 3.763744644916704*^9}}, CellLabel->"In[36]:=",ExpressionUUID->"999c5372-ee2d-4f82-8d19-ec8da8b9497a"], Cell[BoxData["3"], "Output", CellChangeTimes->{3.827494377259925*^9}, CellLabel->"Out[36]=",ExpressionUUID->"b2c7b5ae-493f-4a07-a7df-3c0b923fa77d"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"addOne", "[", RowBox[{"{", RowBox[{"3", ",", "4", ",", "5"}], "}"}], "]"}]], "Input", CellChangeTimes->{{3.7637446458222017`*^9, 3.7637446492798643`*^9}}, CellLabel->"In[37]:=",ExpressionUUID->"fa72b348-9072-456e-a8a9-7a21eb801b36"], Cell[BoxData[ RowBox[{"{", RowBox[{"4", ",", "5", ",", "6"}], "}"}]], "Output", CellChangeTimes->{3.8274944033777313`*^9}, CellLabel->"Out[37]=",ExpressionUUID->"182b50e9-8064-4232-b632-3c0ece2b7a9e"] }, Open ]], Cell["\<\ There are much more sophisticated patterns than just \ \[OpenCurlyDoubleQuote]match any expression.\[CloseCurlyDoubleQuote] We will \ cover these later.\ \>", "Text", CellChangeTimes->{{3.763744651607274*^9, 3.763744681249196*^9}},ExpressionUUID->"28c5134e-0b7d-44a8-8881-\ 640e56b07700"], Cell["\<\ Suppose we want to recursively compute the factorial Base case: fac(0) = 1 Recursive definition: fac(n) = n*fac(n-1)\ \>", "Text", CellChangeTimes->{{3.7637443837940083`*^9, 3.763744428726118*^9}, { 3.7637446855586357`*^9, 3.763744687984498*^9}},ExpressionUUID->"e0999877-e1ee-4507-b62c-\ caaf7239a1cf"], Cell["\<\ This is simple to define in Mathematica. We begin with the base case\ \>", "Text", CellChangeTimes->{{3.763744693191021*^9, 3.763744718756913*^9}},ExpressionUUID->"fcb27bf4-1a46-4230-8eff-\ d653fc16a18d"], Cell[BoxData[ RowBox[{ RowBox[{"fac", "[", "0", "]"}], ":=", "1"}]], "Input", CellChangeTimes->{{3.763744702415689*^9, 3.763744704581748*^9}, 3.763745128927702*^9}, CellLabel->"In[38]:=",ExpressionUUID->"45631687-9ce3-4114-89fa-8a5ae99b7f5d"], Cell["Now, fac[0] will evaluate to 1", "Text", CellChangeTimes->{{3.763744721300274*^9, 3.763744728050556*^9}},ExpressionUUID->"89368bc6-31e1-4ffc-ae30-\ 528efa05a82d"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"fac", "[", "0", "]"}]], "Input", CellChangeTimes->{{3.763744707663625*^9, 3.763744708408098*^9}}, CellLabel->"In[39]:=",ExpressionUUID->"0af131e3-49fe-4705-a368-83c3222eec6b"], Cell[BoxData["1"], "Output", CellChangeTimes->{3.8274944670362873`*^9}, CellLabel->"Out[39]=",ExpressionUUID->"bff634a2-0554-49d1-91c0-9512af339724"] }, Open ]], Cell["\<\ We don\[CloseCurlyQuote]t have a rule for the general case, so fac[n] will \ remain unevaluated:\ \>", "Text", CellChangeTimes->{{3.763744733932872*^9, 3.7637447580607777`*^9}},ExpressionUUID->"57a958a0-d1e6-45b3-813a-\ f19fffb3f677"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"fac", "[", "1", "]"}]], "Input", CellChangeTimes->{{3.763744709936459*^9, 3.763744710729124*^9}}, CellLabel->"In[40]:=",ExpressionUUID->"96d688e7-c2bb-4029-8518-671f4b0c6afe"], Cell[BoxData[ RowBox[{"fac", "[", "1", "]"}]], "Output", CellChangeTimes->{3.827494475513172*^9}, CellLabel->"Out[40]=",ExpressionUUID->"757a15a7-13e1-43b2-9be6-f2cc1d13623e"] }, Open ]], Cell["Defining the general case:", "Text", CellChangeTimes->{{3.76374476124261*^9, 3.763744767426716*^9}},ExpressionUUID->"bd7e641d-30cf-4122-9e91-\ 6838bd0de5b0"], Cell[BoxData[ RowBox[{ RowBox[{"fac", "[", "n_", "]"}], ":=", RowBox[{"n", "*", RowBox[{"fac", "[", RowBox[{"n", "-", "1"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.763744768166182*^9, 3.763744773256482*^9}, 3.763744862831215*^9, 3.7637451404886017`*^9}, CellLabel->"In[41]:=",ExpressionUUID->"fc97b6e5-a5a7-4bad-9ed0-1d755494383a"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"fac", "[", "1", "]"}]], "Input", CellChangeTimes->{{3.7637447749265337`*^9, 3.763744775668865*^9}}, CellLabel->"In[42]:=",ExpressionUUID->"ec941478-bf66-4bab-a0c2-80ba512e08a3"], Cell[BoxData["1"], "Output", CellChangeTimes->{3.8274945360520763`*^9}, CellLabel->"Out[42]=",ExpressionUUID->"9f930f1f-c4ae-4416-a437-0b0aaeb8f11c"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"fac", "[", "10", "]"}]], "Input", CellChangeTimes->{{3.763744776552127*^9, 3.763744777390856*^9}}, CellLabel->"In[43]:=",ExpressionUUID->"2774a4f1-3bb9-451c-871b-1e57e4391679"], Cell[BoxData["3628800"], "Output", CellChangeTimes->{3.8274945568309526`*^9}, CellLabel->"Out[43]=",ExpressionUUID->"d09cb412-b3bb-4714-885b-bf3d36c47653"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"10", "!"}]], "Input", CellChangeTimes->{{3.763744778999363*^9, 3.763744779488647*^9}}, CellLabel->"In[44]:=",ExpressionUUID->"336df37a-a64b-4f71-946c-446f3818ac0f"], Cell[BoxData["3628800"], "Output", CellChangeTimes->{3.827494563430793*^9}, CellLabel->"Out[44]=",ExpressionUUID->"b773db95-ea24-4901-8da3-6d4e6f3c62db"] }, Open ]], Cell[TextData[{ "Let\[CloseCurlyQuote]s do a similar exercise with the Fibonacci sequence:\n", Cell[BoxData[{ FormBox[ RowBox[{ SubscriptBox["F", "0"], "=", "0"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ SubscriptBox["F", "1"], "=", "1"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ SubscriptBox["F", "n"], "=", RowBox[{ SubscriptBox["F", RowBox[{"n", "-", "1"}]], "+", SubscriptBox["F", RowBox[{"n", "-", "2"}]]}]}], TraditionalForm]}],ExpressionUUID-> "c4a96bdb-c9c0-423e-8d50-8045e390d9f0"] }], "Text", CellChangeTimes->{{3.7637447947215643`*^9, 3.7637448454068527`*^9}},ExpressionUUID->"da145968-6f3d-46fc-af7c-\ 4d509398d0df"], Cell[BoxData[{ RowBox[{ RowBox[{"fib", "[", "0", "]"}], ":=", "0"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"fib", "[", "1", "]"}], ":=", "1"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"fib", "[", "n_", "]"}], ":=", RowBox[{ RowBox[{"fib", "[", RowBox[{"n", "-", "1"}], "]"}], "+", RowBox[{"fib", "[", RowBox[{"n", "-", "2"}], "]"}]}]}]}], "Input", CellChangeTimes->{{3.76374484932386*^9, 3.76374486055851*^9}, { 3.7637451623737793`*^9, 3.763745168035763*^9}}, CellLabel->"In[45]:=",ExpressionUUID->"c59e39f2-a594-4889-aafe-6eb329e8d9dd"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"fib", "[", "10", "]"}]], "Input", CellChangeTimes->{{3.7637451691287603`*^9, 3.76374516985737*^9}}, CellLabel->"In[48]:=",ExpressionUUID->"ed7594d3-e19a-4557-b3fe-21ff4600b583"], Cell[BoxData["55"], "Output", CellChangeTimes->{3.827494631662689*^9}, CellLabel->"Out[48]=",ExpressionUUID->"e8b50625-0cc6-4ffe-9bf4-90cef0cf7532"] }, Open ]], Cell["\<\ Of course, Mathematica also has this built in, so we can check our work...\ \>", "Text", CellChangeTimes->{{3.7637451804695683`*^9, 3.763745192676355*^9}},ExpressionUUID->"58e836cb-c5ed-4a18-ae57-\ 3b5351d9fc4f"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Fibonacci", "[", "10", "]"}]], "Input", CellChangeTimes->{{3.763745170970305*^9, 3.7637451761252327`*^9}}, CellLabel->"In[49]:=",ExpressionUUID->"a90be026-a36d-4dc6-96b8-d8ac12b5354c"], Cell[BoxData["55"], "Output", CellChangeTimes->{3.8274946355790653`*^9}, CellLabel->"Out[49]=",ExpressionUUID->"7db97c8b-ef9e-43b3-a1c2-de8a065d97a8"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"e518e880-14b5-4d98-906f-d90da261043f"], Cell[CellGroupData[{ Cell["Solving recurrence relations", "Section", CellChangeTimes->{{3.763745256646412*^9, 3.763745260346157*^9}},ExpressionUUID->"35c742ca-edac-4d97-801a-\ c191c76b5014"], Cell["Suppose we have the recurrence relation", "Text", CellChangeTimes->{{3.763745405011569*^9, 3.763745411414289*^9}},ExpressionUUID->"efefff7f-3836-4aa1-8df9-\ 51cf9ef9cf95"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {GridBox[{ { RowBox[{ RowBox[{ SubscriptBox["a", "0"], "=", "2"}], "\[IndentingNewLine]", RowBox[{ SubscriptBox["a", "1"], "=", "3"}], "\[IndentingNewLine]", RowBox[{ SubscriptBox["a", "n"], "=", RowBox[{ RowBox[{"7", SubscriptBox["a", RowBox[{"n", "-", "1"}]]}], "-", RowBox[{"10", SubscriptBox["a", RowBox[{"n", "-", "2"}]]}]}]}]}]} }, GridBoxAlignment->{"Columns" -> {{"Center"}}}]} }, GridBoxItemSize->{"Columns" -> {{ Scaled[0.96]}}}], TraditionalForm]],ExpressionUUID->"55e4f270-e056-4a9d-8611-87e6c9465dfa"]], \ "Text", CellChangeTimes->{{3.763745472392723*^9, 3.763745490341186*^9}},ExpressionUUID->"57b725e4-c1f4-4645-8399-\ 4c42dec784e4"], Cell["\<\ We can easily create a function to recursively evaluate this sequence\ \>", "Text", CellChangeTimes->{{3.7637454972678432`*^9, 3.7637455111155*^9}},ExpressionUUID->"a9c09f32-a8fb-445e-97cc-b131d545ecae"], Cell[BoxData[{ RowBox[{ RowBox[{"a", "[", "0", "]"}], ":=", "2"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"a", "[", "1", "]"}], ":=", "3"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"a", "[", "n_", "]"}], ":=", RowBox[{ RowBox[{"7", RowBox[{"a", "[", RowBox[{"n", "-", "1"}], "]"}]}], "-", RowBox[{"10", RowBox[{"a", "[", RowBox[{"n", "-", "2"}], "]"}]}]}]}]}], "Input", CellChangeTimes->{{3.763745511934895*^9, 3.7637455318274517`*^9}}, CellLabel->"In[61]:=",ExpressionUUID->"22b0792f-3c94-4f4c-9bba-3258904a0f43"], Cell[BoxData[ RowBox[{"a", "[", "10", "]"}]], "Input", CellChangeTimes->{{3.763745532723859*^9, 3.763745533642973*^9}},ExpressionUUID->"424aeef2-7046-4ca2-ae85-\ 43bfa489c850"], Cell[TextData[{ "What if we want to know a general (non-recursive formula)?\nFirst, \ concentrate on the recursive definition ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["a", "n"], "=", RowBox[{ RowBox[{"7", SubscriptBox["a", RowBox[{"n", "-", "1"}]]}], "-", RowBox[{"10", SubscriptBox["a", RowBox[{"n", "-", "2"}]]}]}]}], TraditionalForm]],ExpressionUUID-> "dfd357c0-29a0-4a4a-b6e1-8452fea2efb1"], " (ignoring initial conditions)\nEquivalently," }], "Text", CellChangeTimes->{{3.7637455364103603`*^9, 3.763745578074292*^9}, { 3.763745883480447*^9, 3.763745904136417*^9}, {3.795878188331449*^9, 3.7958782031963377`*^9}, {3.8272801011556797`*^9, 3.8272801027433023`*^9}},ExpressionUUID->"211ec916-2c2f-4525-9be8-\ a7c6780cbca6"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ { RowBox[{ RowBox[{ SubscriptBox["a", "n"], "-", RowBox[{"7", " ", SubscriptBox["a", RowBox[{"n", "-", "1"}]]}], "+", RowBox[{"10", " ", SubscriptBox["a", RowBox[{"n", "-", "2"}]]}]}], "=", "0"}]} }, GridBoxItemSize->{"Columns" -> {{ Scaled[0.96]}}}], TraditionalForm]],ExpressionUUID->"849dccb0-a1a3-40ec-907c-804b31e0e5cd"]], \ "Text", CellChangeTimes->{{3.795878982249694*^9, 3.795878985770842*^9}},ExpressionUUID->"6b1ccf65-7603-4385-b0f8-\ 0015f8560033"], Cell[TextData[{ "We can solve this recurrence by making the ansatz that ", Cell[BoxData[ FormBox[ SubscriptBox["a", "n"], TraditionalForm]],ExpressionUUID-> "9a728a74-02bc-492d-a339-a52799a6b2fa"], "is given by ", Cell[BoxData[ FormBox[ SuperscriptBox["r", "n"], TraditionalForm]],ExpressionUUID-> "bb3f102f-4317-41fe-ace8-d01bc7f88749"], " for some ", StyleBox["r\n", FontSlant->"Italic"], "In that case," }], "Text", CellChangeTimes->{ 3.795878191529285*^9},ExpressionUUID->"5e917751-249b-40d7-85af-\ 38c92008faa0"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {GridBox[{ { RowBox[{ RowBox[{ SuperscriptBox["r", RowBox[{"n", "-", "2"}]], "(", RowBox[{ SuperscriptBox["r", "2"], "-", RowBox[{"7", "r"}], "+", "10"}], ")"}], "=", "0"}]} }, GridBoxAlignment->{"Columns" -> {{"Center"}}}]} }, GridBoxItemSize->{"Columns" -> {{ Scaled[0.96]}}}], TraditionalForm]],ExpressionUUID->"3c9db534-e537-46b4-a44d-9bd316e16e27"]], \ "Text", CellChangeTimes->{{3.763745584151515*^9, 3.763745608127069*^9}},ExpressionUUID->"eb12f704-9da9-42b2-8fe2-\ 88ff403c7eea"], Cell["So, we look for solutions to this equation.", "Text", CellChangeTimes->{{3.763745612036998*^9, 3.763745627910033*^9}},ExpressionUUID->"bcd8148e-24b6-4d1f-82b1-\ a3980b68cbbc"], Cell[BoxData[ RowBox[{"Factor", "[", RowBox[{ SuperscriptBox["r", "2"], "-", RowBox[{"7", "r"}], "+", "10"}], "]"}]], "Input", CellChangeTimes->{{3.763745628568654*^9, 3.763745647452922*^9}}, CellLabel->"In[56]:=",ExpressionUUID->"b0aa5704-c4d8-4ec6-a01b-7fb5acc52f0b"], Cell[TextData[{ "So, ", Cell[BoxData[ FormBox[ RowBox[{"r", "=", "5"}], TraditionalForm]],ExpressionUUID-> "281e24ef-6ac2-467a-a13b-04ba64fd08c0"], " and ", Cell[BoxData[ FormBox[ RowBox[{"r", "=", "2"}], TraditionalForm]],ExpressionUUID-> "efb57eed-f9ba-46e7-a5a3-c48b289c6dbc"], " satisfy this relationship.\nThe general solution will be given by ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["a", "n"], "=", RowBox[{ RowBox[{"c", "*", SuperscriptBox["5", "n"]}], "+", RowBox[{"d", "*", SuperscriptBox["2", "n"]}]}]}], TraditionalForm]],ExpressionUUID-> "264d7bf7-e233-49f3-a829-64c1660de8d9"] }], "Text", CellChangeTimes->{{3.763745908230575*^9, 3.7637459620259953`*^9}},ExpressionUUID->"993f2bbe-09c9-4f31-a763-\ 6f76db664fa4"], Cell[TextData[{ "We can easily solve the system of equations to find the values of ", StyleBox["c", FontSlant->"Italic"], " and ", StyleBox["d", FontSlant->"Italic"], ".\nBut we can also let Mathematica help..." }], "Text", CellChangeTimes->{{3.763771841765093*^9, 3.7637718441857862`*^9}, { 3.763772704831451*^9, 3.7637727239901247`*^9}},ExpressionUUID->"ba5b2379-78f4-4616-8c71-\ e64e6cdc82a3"], Cell[BoxData[{ RowBox[{"expr", "=", RowBox[{ RowBox[{"c", "*", SuperscriptBox["5", "n"]}], "+", RowBox[{"d", "*", SuperscriptBox["2", "n"]}]}]}], "\[IndentingNewLine]", RowBox[{"case0", "=", RowBox[{"expr", "/.", RowBox[{"n", "\[Rule]", "0"}]}]}], "\[IndentingNewLine]", RowBox[{"case1", "=", RowBox[{"expr", "/.", RowBox[{"n", "\[Rule]", "1"}]}]}]}], "Input", 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Mathematica can also solve the recurrence directly using RSolve\ \>", "Text", CellChangeTimes->{{3.763772964594274*^9, 3.763772971552928*^9}},ExpressionUUID->"ab49b90d-37f2-4f87-9d73-\ 753f3a5234ca"], Cell[BoxData[ RowBox[{"RSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"a", "[", "0", "]"}], "\[Equal]", "2"}], ",", RowBox[{ RowBox[{"a", "[", "1", "]"}], "\[Equal]", "3"}], ",", RowBox[{ RowBox[{"a", "[", "n", "]"}], "==", RowBox[{ RowBox[{"7", RowBox[{"a", "[", RowBox[{"n", "-", "1"}], "]"}]}], "-", RowBox[{"10", RowBox[{"a", "[", RowBox[{"n", "-", "2"}], "]"}]}]}]}]}], "}"}], ",", "a", ",", "n"}], "]"}]], "Input", CellChangeTimes->{{3.763772878184586*^9, 3.7637729233392344`*^9}},ExpressionUUID->"03f8d67f-62e2-4091-bcca-\ 5e50fcfc1fef"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"a85b50a2-cac8-4bb3-80ad-6b8aabfd073e"], Cell[CellGroupData[{ Cell["More complicated functions and local 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