document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210224>Question 1210224</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the number of sequences containing three terms, such that<BR>\n" );
document.write( "* The second term is equal to the sum of the first term plus one.\r<BR>\n" );
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document.write( "* The third term is equal to twice the second term.<BR>\n" );
document.write( "* Each term is an integer in \{0, 1, 2, \dots, 100\}.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851687 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=851687\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the three terms of the sequence be $a_1, a_2, a_3$.<BR>\n" );
document.write( "According to the given conditions:<BR>\n" );
document.write( "\begin{enumerate}<BR>\n" );
document.write( "    \item The second term is equal to the sum of the first term plus one: $a_2 = a_1 + 1$.<BR>\n" );
document.write( "    \item The third term is equal to twice the second term: $a_3 = 2a_2$.<BR>\n" );
document.write( "    \item Each term is an integer in $\{0, 1, 2, \dots, 100\}$.<BR>\n" );
document.write( "\end{enumerate}\r<BR>\n" );
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document.write( "From condition 1, since $a_2 = a_1 + 1$, and $a_2$ is an integer, $a_1$ must also be an integer.<BR>\n" );
document.write( "From condition 2, since $a_3 = 2a_2$, and $a_2$ is an integer, $a_3$ must also be an integer.\r<BR>\n" );
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document.write( "We have the following relationships between the terms:<BR>\n" );
document.write( "$a_2 = a_1 + 1$<BR>\n" );
document.write( "$a_3 = 2(a_1 + 1) = 2a_1 + 2$\r<BR>\n" );
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document.write( "Since each term is in the set $\{0, 1, 2, \dots, 100\}$, we have the following inequalities:<BR>\n" );
document.write( "$0 \le a_1 \le 100$<BR>\n" );
document.write( "$0 \le a_2 \le 100 \implies 0 \le a_1 + 1 \le 100$<BR>\n" );
document.write( "$0 \le a_3 \le 100 \implies 0 \le 2a_1 + 2 \le 100$\r<BR>\n" );
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document.write( "Let's analyze the second inequality:<BR>\n" );
document.write( "$0 \le a_1 + 1 \implies a_1 \ge -1$<BR>\n" );
document.write( "$a_1 + 1 \le 100 \implies a_1 \le 99$<BR>\n" );
document.write( "Combining these, we get $-1 \le a_1 \le 99$.\r<BR>\n" );
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document.write( "Let's analyze the third inequality:<BR>\n" );
document.write( "$0 \le 2a_1 + 2 \implies 2a_1 \ge -2 \implies a_1 \ge -1$<BR>\n" );
document.write( "$2a_1 + 2 \le 100 \implies 2a_1 \le 98 \implies a_1 \le 49$<BR>\n" );
document.write( "Combining these, we get $-1 \le a_1 \le 49$.\r<BR>\n" );
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document.write( "For all three conditions to be satisfied, $a_1$ must satisfy all three inequalities:<BR>\n" );
document.write( "$0 \le a_1 \le 100$<BR>\n" );
document.write( "$-1 \le a_1 \le 99$<BR>\n" );
document.write( "$-1 \le a_1 \le 49$\r<BR>\n" );
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document.write( "The intersection of these three ranges for $a_1$ is $0 \le a_1 \le 49$.<BR>\n" );
document.write( "Since $a_1$ must be an integer, the possible values for $a_1$ are $0, 1, 2, \dots, 49$.\r<BR>\n" );
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document.write( "The number of possible integer values for $a_1$ is $49 - 0 + 1 = 50$.<BR>\n" );
document.write( "Each valid value of $a_1$ uniquely determines the sequence $(a_1, a_1+1, 2(a_1+1))$.\r<BR>\n" );
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document.write( "For example:<BR>\n" );
document.write( "If $a_1 = 0$, the sequence is $(0, 1, 2)$.<BR>\n" );
document.write( "If $a_1 = 49$, the sequence is $(49, 50, 100)$.\r<BR>\n" );
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document.write( "The number of such sequences is equal to the number of possible values for $a_1$, which is 50.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{50}$ </SPAN>\n" );
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