document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209893>Question 1209893</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Compute<BR>\n" );
document.write( "\sum_{k = 1}^{100} k (\lceil \log_3 k \rceil - \lfloor \log_9 k \rfloor). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850982 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=850982\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this summation step-by-step.\r<BR>\n" );
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document.write( "**Understanding the Terms**\r<BR>\n" );
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document.write( "We have the summation:\r<BR>\n" );
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document.write( "$$\sum_{k=1}^{100} k (\lceil \log_3 k \rceil - \lfloor \log_9 k \rfloor)$$\r<BR>\n" );
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document.write( "* **$\lceil \log_3 k \rceil$:** This represents the smallest integer greater than or equal to $\log_3 k$.<BR>\n" );
document.write( "* **$\lfloor \log_9 k \rfloor$:** This represents the largest integer less than or equal to $\log_9 k$.\r<BR>\n" );
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document.write( "**Key Property:**\r<BR>\n" );
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document.write( "We know that $\log_9 k = \log_{3^2} k = \frac{1}{2} \log_3 k$.\r<BR>\n" );
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document.write( "**Analyzing the Difference**\r<BR>\n" );
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document.write( "Let's look at the difference $\lceil \log_3 k \rceil - \lfloor \log_9 k \rfloor$.\r<BR>\n" );
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document.write( "1.  **If $\log_3 k$ is an integer:**<BR>\n" );
document.write( "    * Then $\lceil \log_3 k \rceil = \log_3 k$.<BR>\n" );
document.write( "    * $\lfloor \log_9 k \rfloor = \lfloor \frac{1}{2} \log_3 k \rfloor$.<BR>\n" );
document.write( "    * The difference will be $\log_3 k - \lfloor \frac{1}{2} \log_3 k \rfloor$.\r<BR>\n" );
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document.write( "2.  **If $\log_3 k$ is not an integer:**<BR>\n" );
document.write( "    * We still have $\lfloor \log_9 k \rfloor = \lfloor \frac{1}{2} \log_3 k \rfloor$.\r<BR>\n" );
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document.write( "**Simplifying the Difference**\r<BR>\n" );
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document.write( "Let $x = \log_3 k$. Then we want to understand $\lceil x \rceil - \lfloor x/2 \rfloor$.\r<BR>\n" );
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document.write( "* If $x$ is an integer, then $\lceil x \rceil = x$.<BR>\n" );
document.write( "* If $x$ is not an integer, then $\lceil x \rceil = \lfloor x \rfloor + 1$.\r<BR>\n" );
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document.write( "Let's consider possible values of $\lfloor x/2 \rfloor$.\r<BR>\n" );
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document.write( "* If $x = 2n$, where $n$ is an integer, then $\lfloor x/2 \rfloor = n$, and $\lceil x \rceil = 2n$. The difference is $2n - n = n = x/2$.<BR>\n" );
document.write( "* If $x = 2n + 1$, where $n$ is an integer, then $\lfloor x/2 \rfloor = n$, and $\lceil x \rceil = 2n + 1$. The difference is $2n + 1 - n = n + 1 = (x+1)/2$.<BR>\n" );
document.write( "* If $2n < x < 2n+1$, then $\lfloor x/2 \rfloor = n$. The difference is $\lceil x \rceil - n$.<BR>\n" );
document.write( "* If $2n+1 < x < 2n+2$, then $\lfloor x/2 \rfloor = n$. The difference is $\lceil x \rceil - n$.\r<BR>\n" );
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document.write( "The difference $\lceil \log_3 k \rceil - \lfloor \log_9 k \rfloor$ is 1 if $\log_3 k$ is not an even integer, and is 0 if $\log_3 k$ is an even integer.\r<BR>\n" );
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document.write( "**When is the difference 1?**\r<BR>\n" );
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document.write( "The difference is 1 when $\log_3 k$ is not an even integer. This occurs when $k$ is not a power of $3^2 = 9$.\r<BR>\n" );
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document.write( "**When is the difference 0?**\r<BR>\n" );
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document.write( "The difference is 0 when $\log_3 k$ is an even integer. This occurs when $k$ is a power of $9$.\r<BR>\n" );
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document.write( "**Calculating the Sum**\r<BR>\n" );
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document.write( "We need to calculate $\sum_{k=1}^{100} k (\lceil \log_3 k \rceil - \lfloor \log_9 k \rfloor)$. This is equivalent to summing all $k$ values except for powers of $9$.\r<BR>\n" );
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document.write( "* Powers of 9 in the range: $1, 9, 81$.<BR>\n" );
document.write( "* Sum of all integers from 1 to 100: $\frac{100(101)}{2} = 5050$.<BR>\n" );
document.write( "* Sum of powers of 9: $1 + 9 + 81 = 91$.\r<BR>\n" );
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document.write( "The desired sum is $5050 - 91 = 4959$.\r<BR>\n" );
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document.write( "**Final Answer:** The final answer is $\boxed{4959}$<BR>\n" );
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