document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210222>Question 1210222</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the number of arithmetic sequences such that:<BR>\n" );
document.write( "* The arithmetic sequence contains three terms<BR>\n" );
document.write( "* All the terms are integers in \{0, 1, 2, \dots, 10\}<BR>\n" );
document.write( "* The sum of the terms is 9. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851680 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851680\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the three terms of the arithmetic sequence be $a-d, a, a+d$, where $a$ is the middle term and $d$ is the common difference. Since the terms are integers, $a$ and $d$ must be such that $a-d$ and $a+d$ are also integers.\r<BR>\n" );
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document.write( "The sum of the terms is given as 9:<BR>\n" );
document.write( "$(a-d) + a + (a+d) = 9$<BR>\n" );
document.write( "$3a = 9$<BR>\n" );
document.write( "$a = 3$\r<BR>\n" );
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document.write( "So, the middle term of the arithmetic sequence must be 3.\r<BR>\n" );
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document.write( "The three terms of the sequence are now $3-d, 3, 3+d$.<BR>\n" );
document.write( "We are given that all the terms are in the set $\{0, 1, 2, \dots, 10\}$. This gives us the following inequalities:<BR>\n" );
document.write( "$0 \le 3-d \le 10$<BR>\n" );
document.write( "$0 \le 3 \le 10$ (This is always true)<BR>\n" );
document.write( "$0 \le 3+d \le 10$\r<BR>\n" );
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document.write( "Let's analyze the first inequality:<BR>\n" );
document.write( "$0 \le 3-d \implies d \le 3$<BR>\n" );
document.write( "$3-d \le 10 \implies -7 \le d$<BR>\n" );
document.write( "Combining these, we get $-7 \le d \le 3$.\r<BR>\n" );
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document.write( "Now let's analyze the third inequality:<BR>\n" );
document.write( "$0 \le 3+d \implies -3 \le d$<BR>\n" );
document.write( "$3+d \le 10 \implies d \le 7$<BR>\n" );
document.write( "Combining these, we get $-3 \le d \le 7$.\r<BR>\n" );
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document.write( "For all three terms to be in the set, $d$ must satisfy both $-7 \le d \le 3$ and $-3 \le d \le 7$. The intersection of these two ranges is $-3 \le d \le 3$.\r<BR>\n" );
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document.write( "Since the terms of the arithmetic sequence are integers, $d$ must be an integer. The possible integer values for $d$ are: $-3, -2, -1, 0, 1, 2, 3$.\r<BR>\n" );
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document.write( "Let's list the arithmetic sequences for each value of $d$:<BR>\n" );
document.write( "If $d = -3$: $3-(-3), 3, 3+(-3) \implies 6, 3, 0$<BR>\n" );
document.write( "If $d = -2$: $3-(-2), 3, 3+(-2) \implies 5, 3, 1$<BR>\n" );
document.write( "If $d = -1$: $3-(-1), 3, 3+(-1) \implies 4, 3, 2$<BR>\n" );
document.write( "If $d = 0$: $3-0, 3, 3+0 \implies 3, 3, 3$<BR>\n" );
document.write( "If $d = 1$: $3-1, 3, 3+1 \implies 2, 3, 4$<BR>\n" );
document.write( "If $d = 2$: $3-2, 3, 3+2 \implies 1, 3, 5$<BR>\n" );
document.write( "If $d = 3$: $3-3, 3, 3+3 \implies 0, 3, 6$\r<BR>\n" );
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document.write( "All these sequences contain three terms, all terms are integers in $\{0, 1, 2, \dots, 10\}$, and the sum of the terms is $6+3+0=9$, $5+3+1=9$, $4+3+2=9$, $3+3+3=9$, $2+3+4=9$, $1+3+5=9$, $0+3+6=9$.\r<BR>\n" );
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document.write( "The number of possible integer values for $d$ is $3 - (-3) + 1 = 3 + 3 + 1 = 7$.<BR>\n" );
document.write( "Thus, there are 7 such arithmetic sequences.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{7}$ </SPAN>\n" );
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