Question 1168949
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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.
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<pre>
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.
</pre>

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


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


Simply ignore his post. &nbsp;This &nbsp;Artificial &nbsp;Intelligence is still undertrained.


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