How do i Find the sum of numbers 1-50 without adding them all?

[Polonium 84]

Okay so I saw this [math]\ \sum_{n=1}^{100}[/math] and it was worked out like this:
[math]S= 1+2+3...\\ \underline{S=100+99+98...}\\ S=101+101+101\\ 2S=101(100)\\ S=5050[/math]however it only worked for for 100 my question is how do you work out the sum off all numbers for different limits e.g the sum of all numbers to 49.
P.S I did not know what was the most appropriate to post this .

nvm worked it out

 

[Deleted member 4993]

Okay so I saw this [math]\ \sum_{n=1}^{100}[/math] and it was worked out like this:
[math]S= 1+2+3...\\ \underline{S=100+99+98...}\\ S=101+101+101\\ 2S=101(100)\\ S=5050[/math]however it only worked for for 100 my question is how do you work out the sum off all numbers for different limits e.g the sum of all numbers to 49.
P.S I did not know what was the most appropriate to post this .

nvm worked it out

The above summation is an arithmetic series. study the following:

Arithmetic series formula (video) | Series | Khan Academy

The sum of the first n terms in an arithmetic sequence is (n/2)⋅(a₁+aₙ). It is called the arithmetic series formula. Learn more about it here.

www.khanacademy.org

​

Arithmetic progression - Wikipedia

en.wikipedia.org

Arithmetic Series -- from Wolfram MathWorld​

 

[Steven G]

The sum of the two lines should be 2S, not S, on the left side.

Why not just get your hands dirty and do the same exact process except instead of 100 you put n.

S = 1 + 2 + 3 + 4 + ... + n-1 + n
S = n + n-1 + n-2 + n-3 +.... + 2 + 1
2S=(n+1)+(n+1)+...+(n+1) = n(n+1)

S = ????

 

[balloon Man Mike]

Is it (1+n)*(n/2) ?

 

[Deleted member 4993]

Is it (1+n)*(n/2) ?

The sum of 1+2+3+....n = n * (n+1)/2

You estimated the sum correctly!

 

[BigBeachBanana]

Is it (1+n)*(n/2) ?

This might help you understand the intuition behind the formula. In the picture below, each yellow block represents a unit, and our goal is to find the total number of blocks up to the nth row.

Next, we duplicate the blocks (in green), rotate them 180 degrees and merge them to make a rectangle.
.
Notice that we constructed a rectangle with dimension n*(n+1) by doing so. Ultimately, our goal is to find the area of the yellow triangle, which is half of the rectangle. Thereby,[math]S_n=\frac{n(n+1)}{2}[/math]

 

[Steven G]

The sum of 1+2+3+....n = n * (n+1)/2

You estimated the sum correctly!

Why estimated??

 

[lookagain]

The sum of 1+2+3+....n = n * (n+1)/2

You estimated the sum correctly!

The sum of 1 + 2 + 3 + ... + n = n*(n + 1)/2.

You're missing a fourth plus sign as you count from the left to the right.

Maybe the person "guesstimated" the sum correctly.