Visual Proof: Natural Numbers Formula

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Discover the fascinating world of mathematics with MindSphere! In this maths shorts video, we'll take you on a visual journey to prove the formula for the sum of the first n natural numbers. From the fundamental principles to the stunning revelation, we'll break it down in an easy-to-understand and visually engaging way. Get ready to have your mind blown by the simplicity and beauty of mathematics!
#maths shorts, #MindSphere, mathematics, sum of first n natural numbers, formula, #visual proof
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One more way is also possible
LetS=1+2+3+..+n
First, let’s write S in its original order. Secondly, let’s reverse the order of the terms:



Now, let’s add these two expressions of S term by term:
times

When both sides of the equation are added together
In the expression on the right-hand side, (n+1) appears n times because there are n terms in both the original and reversed sums. So, we can simplify the equation as:
2S=n(n+1)

Dividing both sides by 2, we get:

S=n(n+1)/2

AAlok_Yadav_AA
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This in general also explains triangular numbers, which when represented in the form of dots form a triangle.

projay
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Remember, Gauss created that Formula when he was 7 yrs Old,
when he is adding the natural numbers from 1 to 100

anonymous_FoX
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You can also notice that in, say 1+2+3+4, 1+4 and 2+3 will combine together to make 5 each, and build the formula off of that

skeptical_playz
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The visual proof of 1+4+9+…+n^2 is cool too

Redstoner
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Infinity is created by mathematics
But Mathematics is an infinity 😅
This presentation is very nice such a good example to understand series and sequence, that's why this is my favorite you tube channel . 😊

RohanDey-qs
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The sum of n positive odd integers = (n(n+1))/2

anadiacostadeoliveira
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Why couldn't we say 1/2 n^2 (before adding the other triangle)

ezxd
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this is grade 2 Olympiad maths tho(as a former Olympiad maths almo gold winner)

SeIinaaaa