Mathematical Practices

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Monitoring Progress 1 When looking at sequences, it's a good idea to check both the possibility of an arithmetic and a geometric sequence. To see if a sequence is arithmetic, we look at the differences between consecutive terms by subtracting the first term from the second, then the second from the third, and so on.window.JXQ = window.JXQ || []; window.JXQtable = window.JXQtable || {}; if(!window.JXQtable["Solution19191_1_1776070430_l"]) { window.JXQ.push( function () { var code = function (gid) { var images = []; try {/*board aspects*/ var b=mlg.board([-1.75,5.5,21.25,0.5],{"desktopSize":"medium","style":"usa"}); var el = {};/*vertical segment*/ b.segment(b.node(4.5,0.5),b.node(4.5,5.5),{strokeColor:"gray"});/*default display row headers*/ b.txt(1,3,"\\text{sequence}",{fontSize:1.1}); b.txt(1,5,"\\text{year}",{color:"magenta",fontSize:1.1});/*year*/ b.txt(6,5,"1",{fontSize:1,color:"magenta"}); b.txt(8,5,"2",{fontSize:1,color:"magenta"}); b.txt(10,5,"3",{fontSize:1,color:"magenta"}); b.txt(12,5,"4",{fontSize:1,color:"magenta"}); b.txt(14,5,"5",{fontSize:1,color:"magenta"}); b.txt(16,5,"6",{fontSize:1,color:"magenta"}); b.txt(18,5,"7",{fontSize:1,color:"magenta"}); b.txt(20,5,"8",{fontSize:1,color:"magenta"});/*sequence*/ b.txt(6,3,"50",{fontSize:1}); b.txt(8,3,"80",{fontSize:1}); b.txt(10,3,"128",{fontSize:1}); b.txt(12,3,"205",{fontSize:1}); b.txt(14,3,"328",{fontSize:1}); b.txt(16,3,"524",{fontSize:1}); b.txt(18,3,"839",{fontSize:1}); b.txt(20,3,"1342",{fontSize:1});/*segments*/ for(var i = 6; i
Monitoring Progress 2 Some people are tempted to solve an exercise like this by completing Pascal's Triangle to the tenth row, then adding the terms. However, this sequence of numbers 1,2,4,... appears in a lot of different mathematics, so it is worth taking a look at this sequence as it is. Lets first have a look at the ratios between terms.window.JXQ = window.JXQ || []; window.JXQtable = window.JXQtable || {}; if(!window.JXQtable["Solution19192_1_1061415872_l"]) { window.JXQ.push( function () { var code = function (gid) { var images = []; try {/*board aspects*/ var b=mlg.board([-1.75,5.5,23.25,-1.5],{"desktopSize":"medium","style":"usa"}); var el = {};/*vertical segment*/ b.segment(b.node(4.5,-1.5),b.node(4.5,5.5),{strokeColor:"gray"});/*default display row headers*/ b.txt(1,3,"\\text{sequence}",{fontSize:1.1}); b.txt(1,5,"\\text{row}",{color:"magenta",fontSize:1.1});/*year*/ b.txt(6,5,"1",{fontSize:1,color:"magenta"}); b.txt(8,5,"2",{fontSize:1,color:"magenta"}); b.txt(10,5,"3",{fontSize:1,color:"magenta"}); b.txt(12,5,"4",{fontSize:1,color:"magenta"}); b.txt(14,5,"5",{fontSize:1,color:"magenta"}); b.txt(16,5,"6",{fontSize:1,color:"magenta"}); b.txt(18,5,"7",{fontSize:1,color:"magenta"}); b.txt(20,5,"8",{fontSize:1,color:"magenta"});/*sequence*/ b.txt(6,3,"1",{fontSize:1}); b.txt(8,3,"2",{fontSize:1}); b.txt(10,3,"4",{fontSize:1}); b.txt(12,3,"8",{fontSize:1}); b.txt(14,3,"16",{fontSize:1}); b.txt(16,3,"32",{fontSize:1}); b.txt(18,3,"64",{fontSize:1}); b.txt(20,3,"128",{fontSize:1});/*segments*/ for(var i = 6; i