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 February 18th, 2015, 06:26 PM #1 Newbie   Joined: Feb 2015 From: Lucknow, India Posts: 4 Thanks: 0 Am I correct.? Okay, so I was just solving my book exercises and I figured out something. To explain it: Let us take two continuous numbers (like 2 and 3) x and y where x 9 - 4= 5. 9^2 - 8^2 = 2X8+1 -> 81 - 64 = 17 Is this equation new or it was already made by somebody?? Is this of any use to mathematics?? Last edited by skipjack; February 18th, 2015 at 09:21 PM.
February 18th, 2015, 07:18 PM   #2
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Quote:
 Originally Posted by abhigyan001 In simple words, the square of larger number - the square of smaller number will be equal to twice the smaller number +1.
$\displaystyle y-x = 1 \Longrightarrow \ y=x+1\ \Longrightarrow\ y^2=(x+1)^2 \\\;\\ y^2-x^2=(x+1)^2-x^2=x^2+2x+1-x^2=2x+1.$

Last edited by skipjack; February 18th, 2015 at 09:21 PM.

 February 18th, 2015, 09:29 PM #3 Math Team   Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408 Hello, abhigyan001! You have "discovered" that the difference of two consecutive squares is an odd number. $\;\;\;\;\begin{array}{ccc}1^2\,-\,0^2 &=& 1 \\ 2^2\,-\,1^2 &=& 3 \\ 3^2\,-\,2^2 &=& 5 \\ 4^2\,-\,3^2 &=& 7 \\ \vdots && \vdots \end{array}$ $\text{Proof: }\:(x\,+\,1)^2\,-\,x^2 \;=\; x^2\,+\,2x\,+\,1\,-\,x^2 \;=\;2x\,+\,1$ I've used this fact to do some mental arithmetic. We know that $20^2= 400$ What is the next square? $\,21^2$ $\text{Double the 20 and add 1: }\,2(20)\,+\,1 \:=\:41$ $\text{Add this to the previous square: }\,400\,+\,41 \:=\:441$ $\text{Therefore: }\:21^2 \:=\:441$ What is the next square? $\,22^2$ Consecutive squares differ by consecutive odd numbers. The next odd number is 43. $\text{Add this to the previous square: }\,441\,+\,43 \:=\:484$ $\text{Therefore: }\:22^2 \:=\:484$ What is the next square? $\,23^2$ The next odd number is 45. $\text{Then we have: \:484\,+\,45 \:=\:529$ $\text{Therefore: }\:23^2 \:=\:529$ So you can quickly make a list of consecutive squares $\;\;\;$without doing all that multiplication.
 February 18th, 2015, 09:29 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,926 Thanks: 2205 y - x = 1 implies y + x = 2x + 1, and multiplying these equations gives yÂ² - xÂ² = 2x + 1.
 February 18th, 2015, 09:49 PM #5 Math Team   Joined: Apr 2012 Posts: 1,579 Thanks: 22 (x+1)^2 = x^2+2x+1 You can also see it "physically" by considering what you would do to make an n by n square of tiles (or whatever) into an n+1 by n+1 square. You would add n tiles along the bottom, n tiles along one side and one tile in the new corner. Nice that you noticed this pattern on your own. Keep seeking these patterns and you will have a deeper understanding of numbers than you would have by just applying the standard rules of math by rote. I suspect all number theorists start out as kids who notice patterns that show that numbers and all of math are more intriguing than you'd guess just by learning the rules being taught. Last edited by skipjack; February 18th, 2015 at 09:53 PM.

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