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i need help to solve a question that say :

Sn the sum of the first n odd numbers

1 - Formalize this sum: (Give mathematical expression)?

2 - Calculate this sum?

how to solve that ? :/

Adel
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3 Answers3

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HINT

  • consider the sum of the first $2n$ numbers, that is

$$\sum_{k=1}^{2n} k=1+2+3+\ldots+(2n-1)+2n$$

  • subtract the sum of the first $n$ even numbers picking out a factor 2

$$2+4+6+\ldots+(2n-2)+2n=2\sum_{k=1}^{n} k$$

user
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  • i cant understand, can u explain ! – Adel Jul 31 '18 at 22:06
  • The idea is to sum all the even and odd numbers since we know the formula, that is for example to obtain the sum of the first 3 odd numbers, that is up to 5 we calculate: $$1+3+5=(1+2+3+4+5+6)-(2+4+6)=(1+2+3+4+5+6)-2(1+2+3)$$ Can you see the method now? – user Jul 31 '18 at 22:09
  • yes, now its very clear thank u :) – Adel Jul 31 '18 at 22:10
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    @Adel Well done. You are welcome! If you want to show your final expression here above in the OP, I can get a look to it. Bye – user Jul 31 '18 at 22:11
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There is a very neat pictorial proof of the solution dated as far back as the Pythagorean schools who considered every number to have a shape apparently.

Try drawing the odd numbers as "L" shapes of dots,

i.e. \begin{array}{cc} \blacksquare \\ \\ \end{array} \begin{array}{cc} \blacksquare \\ \blacksquare & \blacksquare\\ \end{array} \begin{array}{cc} \blacksquare \\ \blacksquare \\ \blacksquare & \blacksquare & \blacksquare\\ \end{array}for 1, 3, and 5 and so on, then see what happens when you put them together.

p.s. apologies for my LaTeX, if someone can make it look prettier by all means go ahead and edit. Thanks

user
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YeatsL
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  • +1: good answer. Suggestion about the $\LaTeX$: \color seems to work with MathJax {\color{blue}\blacksquare} etc. gives me: $\color{blue}\blacksquare$, $\color{red}\blacksquare$, $\color{green}\blacksquare$, $\ldots$ – Rob Arthan Jul 31 '18 at 22:37
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Here's the $A=B$ solution: $S_1 = 1 = 1 = 1^2$, $S_2 = 1 + 3 = 4 = 2^2$, $S_3 = 1 + 3 + 5 = 9 = 3^2$. Taking differences, we see that the formula must be a quadratic in $n$; three values of a quadratic determine the quadratic: we must have $S_n = n^2$ for all $n$.

Rob Arthan
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