SOLUTION: A quilt is made up of strips of cloth, starting with an inner square surrounded by rectangles to form successively larger squares. The inner square and all rectangles have a wi

Question 1168949: A quilt is made up of strips of
cloth, starting with an inner square surrounded
by rectangles to form successively larger
squares. The inner square and all rectangles
have a width of 1 foot. Write an expression
using summation notation that gives the sum of
the areas of all the strips of cloth used to make
the quilt shown. Then evaluate the expression.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's analyze the pattern of the quilt construction:
* **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.
* Area of Strip 1 = $1 \times 1 = 1$ square foot.
* **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).
* Area of each rectangle = $1 \times 1 = 1$ square foot.
* Total area of Strip 2 = $4 \times 1 = 4$ square feet.
* **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).
* Area of each rectangle = $1 \times 3 = 3$ square feet.
* Total area of Strip 3 = $4 \times 3 = 12$ square feet.
* **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).
* Area of each rectangle = $1 \times 5 = 5$ square feet.
* Total area of Strip 4 = $4 \times 5 = 20$ square feet.
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 $2n - 1$.
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 $4(2n - 3)$.
Let's re-examine the pattern of the total area of each strip:
Strip 1: 1
Strip 2: 4
Strip 3: 12
Strip 4: 20
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 $1 \times s$.
Let $s_n$ be the side length of the square after adding the $n$-th strip.
$s_0 = 1$ (inner square)
$s_1 = 3$
$s_2 = 5$
$s_n = 2n + 1$ (if we consider the number of added strips)
Let's consider the strips.
Strip 1 (n=1): Area = $1^2 = 1$.
Strip 2 (n=2): Adds around a $1 \times 1$ square. Four rectangles of $1 \times 1$. Area = $4 \times 1 = 4$.
Strip 3 (n=3): Adds around a $3 \times 3$ square. Four rectangles of $1 \times 3$. Area = $4 \times 3 = 12$.
Strip 4 (n=4): Adds around a $5 \times 5$ square. Four rectangles of $1 \times 5$. Area = $4 \times 5 = 20$.
Strip $n$ (for $n > 1$): Adds around a $(2n-3) \times (2n-3)$ square. Four rectangles of $1 \times (2n-3)$. Area = $4(2n-3)$.
Let $A_i$ be the area of the $i$-th strip.
$A_1 = 1$
$A_i = 4(2i - 3)$ for $i > 1$.
Suppose the quilt shown has $k$ strips. The sum of the areas of all the strips is:
Sum $= A_1 + \sum_{i=2}^{k} A_i = 1 + \sum_{i=2}^{k} 4(2i - 3)$
Let's assume the quilt shown has 4 strips (based on our initial analysis).
Sum $= 1 + \sum_{i=2}^{4} 4(2i - 3) = 1 + [4(2(2) - 3) + 4(2(3) - 3) + 4(2(4) - 3)]$
Sum $= 1 + [4(4 - 3) + 4(6 - 3) + 4(8 - 3)]$
Sum $= 1 + [4(1) + 4(3) + 4(5)]$
Sum $= 1 + [4 + 12 + 20] = 1 + 36 = 37$ square feet.
Now let's write the expression using summation notation for a quilt with $k$ strips:
Sum $= 1 + \sum_{i=2}^{k} 4(2i - 3)$
To evaluate this expression for $k=4$:
Sum $= 1 + 4 \sum_{i=2}^{4} (2i - 3) = 1 + 4 [(2(2) - 3) + (2(3) - 3) + (2(4) - 3)]$
Sum $= 1 + 4 [(4 - 3) + (6 - 3) + (8 - 3)]$
Sum $= 1 + 4 [1 + 3 + 5] = 1 + 4 [9] = 1 + 36 = 37$.
Final Answer: The final answer is $\boxed{\text{Expression: } 1 + \sum_{i=2}^{k} 4(2i - 3), \text{ Evaluation (for 4 strips): } 37}$
Answer by ikleyn(52781) (Show Source): You can put this solution on YOUR website!
.
A quilt is made up of strips of cloth, starting with an inner square
surrounded by rectangles to form successively larger squares.
The inner square and all rectangles have a width of 1 foot.
Write an expression using summation notation that gives the sum of
the areas of all the strips of cloth used to make the quilt shown.
Then evaluate the expression.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Step 1. At step 1, we have the inner 1x1-square in the center. It is surrounded by 4 (four) rectangles of the length 2 ft and the width 1 ft. These 5 shapes form a 3x3-square with the area 3*3 = 9 square feet. At this point, we can write this equality 1 + 4*2 = 9 square feet. Step 2. At step 2, we have this 3x3-square surrounded by 4 (four) rectangles of the length 4 ft and the width 1 ft. Altogether, they form 5x5-square with the area 5*5 = 25 square feet. At this point, we can write this equality 1 + 4*2 + 4*4 = 25 square feet. Step 3. At step 3, we have this 5x5-square surrounded by 4 (four) rectangles of the length 6 ft and the width 1 ft. Altogether, they form 7x7-square with the area 7*7 = 49 square feet. At this point, we can write this equality 1 + 4*2 + 4*4 + 4*6 = 49 square feet. Step 4. At step 4, we have this 7x7-square surrounded by 4 (four) rectangles of the length 8 ft and the width 1 ft. Altogether, they form 9x9-square with the area 9*9 = 81 square feet. At this point, we can write this equality 1 + 4*2 + 4*4 + 4*6 + 4*8 = 81 square feet. . . . . . and so on . . . . . The pattern is just seen. For clarity, I will describe the common step 'n'. Step n. At step n, we have (2n-1) x (2n-1)-square from the previous step, surrounded by 4 (four) rectangles of the length 2n ft and the width 1 ft. Altogether, they form (2n+1) x (2n+1)-square with the area (2n+1)*(2n+1) = (2n+1)^2 square feet. At this point, we can write this equality 1 + 4*2 + 4*4 + 4*6 + 4*8 + . . . + 4(2n) = (2n+1)^2. It can be proved by the method of Mathematical induction. The step of induction is this Prove that (2n-1)^2 + 4*(2n) = (2n+1)^2. To prove it, simply open parentheses in the left side and simplify (2n-1)^2 + 4*(2n) = 4n^2 - 2*(2n) + 1 + 4*(2n) = 4n^2 + 2*(2n) + 1 = (2n+1)^2. So, the proof is in one line. It is obvious at the same degree, as the geometric step-by-step procedure described above in my post.

At this point, all explanations are complete and the problem is solved in full.

--------------------

Tutor @HPhill (which represents and uses an Artificial Intelligence)
did not get understanding the problem and shot out the target to NOWHERE.

Simply ignore his post. This Artificial Intelligence is still undertrained.

It works perfectly, it it finds a source in its database to re-write from,
but can not think independently.

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