document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1010974>Question 1010974</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find both an explicit formula and recursive formula for the nth term for arithmetic sequence 76,70,63 </SPAN>\n" );
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document.write( "Question:<BR>\n" );
document.write( "Find both an explicit formula and recursive formula for the nth term for arithmetic sequence 76,70,63<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "The given sequence is NOT an arithmetic since the difference increases from 6 to seven.<BR>\n" );
document.write( "Assuming the sequence is a quadratic function.\r<BR>\n" );
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document.write( "A. Recursive formula<BR>\n" );
document.write( "Recall that the difference increases by one for successive terms, the recursive formula is of the form:<BR>\n" );
document.write( "T(n)=T(n-1)-(n+k),<BR>\n" );
document.write( "Since T(1)=76, T(2)=T(1)-(n+k)=> 70=76-(2+k) => k=4<BR>\n" );
document.write( "therefore<BR>\n" );
document.write( "T(n)=T(n-1)-(n+4), with T(1)=76<BR>\n" );
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document.write( "Check: T(2)=76-(2+4)=70; T(3)=70-(3+4)=63, ok.<BR>\n" );
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document.write( "B. Explicit formula:<BR>\n" );
document.write( "We have only three known terms, and knowing that the sequence is not arithmetic, we will assume the sequence is quadratic, of the form:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%28n%29=a%2An%5E2%2Bb%2An%2Bc ALIGN=MIDDLE  ALT=\"T%28n%29=a%2An%5E2%2Bb%2An%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "from which we can substitute for n=1, 2 and 3 to get<BR>\n" );
document.write( "76=a+b+c<BR>\n" );
document.write( "70=4a+2b+c<BR>\n" );
document.write( "63=9a+3b+c<BR>\n" );
document.write( "from which we can readily solve (by elimination) to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-1%2F2 ALIGN=MIDDLE  ALT=\"a=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-9%2F2 ALIGN=MIDDLE  ALT=\"b=-9%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=81 ALIGN=MIDDLE  ALT=\"c=81\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Hence the explicit formula for T(n) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%28n%29=-0.5%2An%5E2+-+4.5%2An+%2B+81 ALIGN=MIDDLE  ALT=\"T%28n%29=-0.5%2An%5E2+-+4.5%2An+%2B+81\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Check:<BR>\n" );
document.write( "T(1)=-0.5-4.5+81=76<BR>\n" );
document.write( "T(2)=-2-9+81=70<BR>\n" );
document.write( "T(3)=-4.5-13.5+81=63<BR>\n" );
document.write( "All satisfied. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );