Linked Questions
84 questions linked to/from Proof $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
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 ...