Linked Questions

12
votes
8answers
2k views

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 $2x + 1$ ...
2
votes
3answers
1k views

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 ...
2
votes
1answer
3k views

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: $n=1, LHS: 2(1)-1 = 1, RHS = 1^2 = 1, True$ ...
-1
votes
4answers
682 views

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 ...
1
vote
3answers
274 views

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 ...
1
vote
3answers
127 views

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?
1
vote
1answer
134 views

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 (“. ....
0
votes
1answer
181 views

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
6
votes
5answers
2k views

Difficulties in a proof by mathematical induction (involves evaluating $\sum r3^r$).

Please help. I've been stuck on this for 2 days. Haven't found any easy explaining text. The question is: Prove by mathematical induction that : $$ \sum_{r=1}^n r3^r = \frac{3}{4} \left[ 3^n \...
9
votes
2answers
387 views

Seeking a combinatorial proof of the identity$1+3+\cdots+(2n-1)=n^2$ [closed]

I would appreciate if somebody could help me with the following problem Q: Seeking a combinatorial proof $$1+3+\cdots+(2n-1)=n^2$$
4
votes
4answers
308 views

Prove for all $m, n \in \mathbb N$: $[1 + 3 +\cdots + (2n -1)]^m = n^{2m}$

I have this: Prove for all $m, n \in \mathbb N$: $$[1 + 3 + \cdots + (2n - 1)]^m = n^{2m}$$ For $n = 1: 1 = 1^2$, hence P(1) is true. Let $N \in \mathbb N$ be given and assume: $$[1 + 3 + \cdots +...
1
vote
2answers
175 views

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 ...
0
votes
4answers
67 views

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 ...
0
votes
2answers
173 views

Prove the theorem of Nicomachus by induction

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 ...
2
votes
1answer
79 views

help deriving a closed formula for this “magic function”

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(...