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### I found this odd relationship, $x^2 = \sum_\limits{k = 0}^{x-1} (2k + 1)$. [duplicate]

I stumbled across this relationship while I was messing around. What's the proof, and how do I understand it intuitively? It doesn't really make sense to me that the sum of odd numbers up to $2x + 1$ ...
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### interpret a sum geometrically [duplicate]

I have this sum: $1+3+5+...+(2n+1)$, where $n$ is a natural number. I have to calculate it and interpret geometrically. Well, it's easy to find out that it equals $(n+1)^2$. But how to interpret it ...
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### Proof By Induction - $n^2 = \sum_{i=1} ^{n} (2i-1)$ for all $n\geq 1$ [duplicate]

Using Proof By Induction I am trying to prove the following: $n^2 = \sum_{i=1} ^{n} (2i-1)$ for all $n\geq 1$ Here is my solutions so Far: Base Case: $n=1, LHS: 2(1)-1 = 1, RHS = 1^2 = 1, True$ ...
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### Formula for finding sequence of odd numbers starting from 0? [duplicate]

Let's say I had an odd number sequence: $$1+3+5+7+9+...+n=x$$ I was doing some problems and I coincidentally thought of: $x=(\frac{n+1}{2})^2$ I searched up the formula online to see if it actually ...
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### Why does summing the first $n$ odd integers give $n^2$? [duplicate]

In a blog post I see a multiplication chart: The post then says: ...can you see why summing the first $n$ odd integers results in $n^2$? But I can't see it. Can someone help me understand why ...
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### How does $\lim\limits_{n\to\infty} \sum_{i=1}^{n^2} 1$ result into 1+3+5… [duplicate]

How can $\lim\limits_{n\to\infty} \sum_{i=1}^{n^2} 1$ result into 1+3+5...? Why odd numbers?
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### How to find closed formula for the sum of the first n odd perfect squares? [duplicate]

I do know formula for finding sum of squares of n odd numbers but I am not sure how to find closed formula. Note: From closed formula I mean a reduced fraction in factored form, with no ellipses (“. ....
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### Summation notation of X^2 [duplicate]

Is this correct? If so a link to more information/proof would be appreciated. Thanks! $$\sum_{k=1}^x(k + k - 1) = x^2$$ WolframAlpha
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### Pattern for square numbers

Due apologies for this rustic image. But while drawing this lattice arrangement about the "square numbers" , I discovered a pattern here wherein if I add the alternate red dots (as depicted in the ...
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### How to find 50-th term of this sequence and its sum

Can someone help me with this excercise? I recognize that the numerator is simply increasing by factors of 2, and every time a new number is written it begins from 1 again. I know that the ...
Prove the theorem of Nicomachus(AD.100) by induction: $$1^3 = 1,\ 2^3 = 3+ 5,\ 3^3 = 7 + 9 + 11,\ 4^3 = 13 + 15 + 17 + 19,\ ...$$ My approach: from looking at the above pattern you can tell ...
I'm having trouble coming up with a closed formula for $n$ from the sequence of numbers generated by this function: The following mystery function $M : N \times N \rightarrow N$ is defined by:  M(...