document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168949>Question 1168949</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A quilt is made up of strips of<BR>\n" );
document.write( "cloth, starting with an inner square surrounded<BR>\n" );
document.write( "by rectangles to form successively larger<BR>\n" );
document.write( "squares. The inner square and all rectangles<BR>\n" );
document.write( "have a width of 1 foot. Write an expression<BR>\n" );
document.write( "using summation notation that gives the sum of<BR>\n" );
document.write( "the areas of all the strips of cloth used to make<BR>\n" );
document.write( "the quilt shown. Then evaluate the expression. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851465 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=851465\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze the pattern of the quilt construction:\r<BR>\n" );
document.write( "\n" );
document.write( "* **Strip 1 (Inner Square):** This is a square with a width of 1 foot. Since it's a square, its length is also 1 foot.<BR>\n" );
document.write( "    * Area of Strip 1 = $1 \times 1 = 1$ square foot.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Strip 2 (First set of rectangles):** This strip surrounds the inner square. It consists of four rectangles, each with a width of 1 foot. The length of each rectangle is equal to the side length of the inner square (1 foot).<BR>\n" );
document.write( "    * Area of each rectangle = $1 \times 1 = 1$ square foot.<BR>\n" );
document.write( "    * Total area of Strip 2 = $4 \times 1 = 4$ square feet.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Strip 3 (Second set of rectangles):** This strip surrounds the shape formed by the inner square and the first strip of rectangles. The side length of this new square is $1 + 1 + 1 = 3$ feet. This strip also consists of four rectangles with a width of 1 foot. The length of each rectangle is equal to the side length of the square it's being added to (3 feet).<BR>\n" );
document.write( "    * Area of each rectangle = $1 \times 3 = 3$ square feet.<BR>\n" );
document.write( "    * Total area of Strip 3 = $4 \times 3 = 12$ square feet.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Strip 4 (Third set of rectangles):** This strip surrounds the shape formed by the previous strips. The side length of this new square is $3 + 1 + 1 = 5$ feet. This strip consists of four rectangles with a width of 1 foot and a length equal to the side length of the square it's being added to (5 feet).<BR>\n" );
document.write( "    * Area of each rectangle = $1 \times 5 = 5$ square feet.<BR>\n" );
document.write( "    * Total area of Strip 4 = $4 \times 5 = 20$ square feet.\r<BR>\n" );
document.write( "\n" );
document.write( "We can see a pattern here. Let $n$ be the strip number (starting with the inner square as $n=1$). The side length of the square just before adding the $n$-th strip can be expressed as $2(n-1) - 1 + 2 = 2n - 2 + 1 = 2n - 1$ for $n > 1$, and 1 for $n=1$. However, it's easier to see the sequence of side lengths of the squares: 1, 3, 5, ... which is $2n - 1$.\r<BR>\n" );
document.write( "\n" );
document.write( "The length of the rectangles in the $n$-th strip (for $n > 1$) is the side length of the square formed by the first $n-1$ strips, which is $2(n-1) - 1 = 2n - 3$. The width is always 1. There are 4 such rectangles. So the area of the $n$-th strip (for $n > 1$) is $4(2n - 3)$.\r<BR>\n" );
document.write( "\n" );
document.write( "Let's re-examine the pattern of the total area of each strip:<BR>\n" );
document.write( "Strip 1: 1<BR>\n" );
document.write( "Strip 2: 4<BR>\n" );
document.write( "Strip 3: 12<BR>\n" );
document.write( "Strip 4: 20\r<BR>\n" );
document.write( "\n" );
document.write( "This doesn't directly fit $4(2n-3)$. Let's think about the increase in area at each step. When we add a strip of width 1 around a square of side length $s$, we add four rectangles of size $1 \times s$.\r<BR>\n" );
document.write( "\n" );
document.write( "Let $s_n$ be the side length of the square after adding the $n$-th strip.<BR>\n" );
document.write( "$s_0 = 1$ (inner square)<BR>\n" );
document.write( "$s_1 = 3$<BR>\n" );
document.write( "$s_2 = 5$<BR>\n" );
document.write( "$s_n = 2n + 1$ (if we consider the number of added strips)\r<BR>\n" );
document.write( "\n" );
document.write( "Let's consider the strips.<BR>\n" );
document.write( "Strip 1 (n=1): Area = $1^2 = 1$.<BR>\n" );
document.write( "Strip 2 (n=2): Adds around a $1 \times 1$ square. Four rectangles of $1 \times 1$. Area = $4 \times 1 = 4$.<BR>\n" );
document.write( "Strip 3 (n=3): Adds around a $3 \times 3$ square. Four rectangles of $1 \times 3$. Area = $4 \times 3 = 12$.<BR>\n" );
document.write( "Strip 4 (n=4): Adds around a $5 \times 5$ square. Four rectangles of $1 \times 5$. Area = $4 \times 5 = 20$.<BR>\n" );
document.write( "Strip $n$ (for $n > 1$): Adds around a $(2n-3) \times (2n-3)$ square. Four rectangles of $1 \times (2n-3)$. Area = $4(2n-3)$.\r<BR>\n" );
document.write( "\n" );
document.write( "Let $A_i$ be the area of the $i$-th strip.<BR>\n" );
document.write( "$A_1 = 1$<BR>\n" );
document.write( "$A_i = 4(2i - 3)$ for $i > 1$.\r<BR>\n" );
document.write( "\n" );
document.write( "Suppose the quilt shown has $k$ strips. The sum of the areas of all the strips is:<BR>\n" );
document.write( "Sum $= A_1 + \sum_{i=2}^{k} A_i = 1 + \sum_{i=2}^{k} 4(2i - 3)$\r<BR>\n" );
document.write( "\n" );
document.write( "Let's assume the quilt shown has 4 strips (based on our initial analysis).<BR>\n" );
document.write( "Sum $= 1 + \sum_{i=2}^{4} 4(2i - 3) = 1 + [4(2(2) - 3) + 4(2(3) - 3) + 4(2(4) - 3)]$<BR>\n" );
document.write( "Sum $= 1 + [4(4 - 3) + 4(6 - 3) + 4(8 - 3)]$<BR>\n" );
document.write( "Sum $= 1 + [4(1) + 4(3) + 4(5)]$<BR>\n" );
document.write( "Sum $= 1 + [4 + 12 + 20] = 1 + 36 = 37$ square feet.\r<BR>\n" );
document.write( "\n" );
document.write( "Now let's write the expression using summation notation for a quilt with $k$ strips:<BR>\n" );
document.write( "Sum $= 1 + \sum_{i=2}^{k} 4(2i - 3)$\r<BR>\n" );
document.write( "\n" );
document.write( "To evaluate this expression for $k=4$:<BR>\n" );
document.write( "Sum $= 1 + 4 \sum_{i=2}^{4} (2i - 3) = 1 + 4 [(2(2) - 3) + (2(3) - 3) + (2(4) - 3)]$<BR>\n" );
document.write( "Sum $= 1 + 4 [(4 - 3) + (6 - 3) + (8 - 3)]$<BR>\n" );
document.write( "Sum $= 1 + 4 [1 + 3 + 5] = 1 + 4 [9] = 1 + 36 = 37$.\r<BR>\n" );
document.write( "\n" );
document.write( "Final Answer: The final answer is $\boxed{\text{Expression: } 1 + \sum_{i=2}^{k} 4(2i - 3), \text{ Evaluation (for 4 strips): } 37}$ </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );