Linked Questions

80
votes
4answers
71k views

What is the term for a factorial type operation, but with summation instead of products? [duplicate]

(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems) I'm aware of Sigma notation, but is there a function/name ...
9
votes
9answers
120k views

What's the formula for the 365 day penny challenge? [duplicate]

Not exactly a duplicate since this is answering a specific instance popular in social media. You might have seen the viral posts about "save a penny a day for a year and make $667.95!" The ...
14
votes
2answers
65k views

Sum of n consecutive numbers [duplicate]

Possible Duplicate: Proof for formula for sum of sequence $1+2+3+\ldots+n$? Is there a shortcut method to working out the sum of n consecutive positive integers? Firstly, starting at $1 ... 1 + ...
2
votes
6answers
257 views

What must be the simplest proof of the sum of first $n$ natural numbers? [duplicate]

I was studying sequence and series and used the formula many times $$1+2+3+\cdots +n=\frac{n(n+1)}{2}$$ I want its proof. Thanks for any help.
5
votes
4answers
16k views

Proving the sum of the first $n$ natural numbers by induction [duplicate]

I am currently studying proving by induction but I am faced with a problem. I need to solve by induction the following question. $$1+2+3+\ldots+n=\frac{1}{2}n(n+1)$$ for all $n > 1$. Any ...
2
votes
5answers
583 views

Using Direct Proof. $1+2+3+\ldots+n = \frac{n(n + 1)}{2}$ [duplicate]

I need help proving this statement. Any help would be great!
3
votes
5answers
486 views

Can you explain this please $T(n) = (n-1)+(n-2)+…1= \frac{(n-1)n}{2}$ [duplicate]

Possible Duplicate: Proof for formula for sum of sequence 1+2+3+…+n? Can you explain this please $$T(n) = (n-1)+(n-2)+…1= \frac{(n-1)n}{2}$$ I am really bad at maths but need to ...
3
votes
5answers
149 views

Sum of the first natural numbers: how many and what are the most common methods to verify it? [duplicate]

We know that Gauss has shown that the sum $S$ of the first $n$ natural numbers is given by the relation: $$S=\frac{n(n+1)}{2} \tag{*}$$ The proof that I remember most frequently is as follows: Let ...
1
vote
3answers
164 views

Why does $1+2+\dots+2003=\dfrac{2004\cdot2003}2$? [duplicate]

Why does $1+2+\dots+2003=\dfrac{2004\cdot2003}2$? Sorry if this is missing context; not really much to add...
0
votes
4answers
195 views

Find the sum $\sum_{j=0}^{n}j$ [duplicate]

Find the sum $\sum_{j=0}^{n}j$ for all $n=0,1,2,3,\dots$. How do I find/evaluate this sum?
0
votes
1answer
2k views

How do I expand/solve the following summation? [duplicate]

$\sum\limits_{i=1}^{n-1} i$. I know the answer is $\frac{1}{2}(n-1)n$ but I don't quite understand it how to get there.
-2
votes
4answers
145 views

Evaluate the sum $1+2+3+…+n$ [duplicate]

How do we evaluate the sum: \begin{equation*} 1+2+...+n \end{equation*} I don't need the proof with the mathematical induction, but the technique to evaluate this series.
2
votes
1answer
200 views

How does this image prove the identity $1+2+3+4\cdots + (n-1) = \binom{n}{2}$? [duplicate]

Possible Duplicate: Proof for formula for sum of sequence 1+2+3+…+n? Proof without words: $\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad $ How does ...
0
votes
2answers
52 views

Prove $\frac{n(n+1)}{2}$ by induction, triangular numbers [duplicate]

Prove that the $n$-th triangular number is: $$T_n = \dfrac{n(n+1)}{2}$$ I did this: Base case: $\frac{1(1+1)}{2}=1$, which is true. Then I assumed that $T_k=\frac{k(k+1)}{2}$ is true. $$T_{k+...
1
vote
6answers
137 views

Why does $ 1+2+3+\cdots+p = {(1⁄2)}\cdots(p+1) $ [duplicate]

I saw this from Project Euler, problem #1: If we now also note that $ 1+2+3+\cdots+p = {(1/2)} \cdot p\cdot(p+1) $ What is the intuitive explanation for this? How would I go about deriving the ...

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