The sum of the first n natural numbers
What is the sum of the first n natural numbers?
Let's look at this problem for n=1, 2, 3, 4, and 5 and calculate the sum:
1=1
1+2=3
1+2+3=6
1+2+3+4=10
1+2+3+4+5=15
What is the formula to calculate this sum:
The answer is
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Related sums:
The sum of the first n odd natural numbers
The sum of the squares of the first n natural numbers
Here is a calculator that calculates this function for you:
n:
1+2+...+n:
We shall give three different proofs for this formula.
Proof 1:
This is an example for n = 5. We see that we have a big rectangle with the its sides 5 and 5+1. The rectangle has 2(1 + 2 + 3 + 4 + 5) squares inside. So 2(1 + 2+ 3 + 4 + 5) = 5(5+1) and 1 + 2 + 3 + 4 + 5 =
The same way, for any n, we construct a rectangle with its sides n and n+1 that has 2(1+2+...+n) squares inside.
Proof 2:
(1+2+....+(n-1)+n) +
(n+(n-1)+....+2+1) =
if we look at the sum above we notice that we have n columns of numbers and that on each column we have two numbers with the sum n+1, so:
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Proof 3:
Let's write like this: ,
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. It's shorter.
We will use induction to prove that:
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1) For n = 1 it is true that
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2)We know that
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is true and we want to prove it for n+1.
We have to prove that:
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if we add n+1 to each side of the next identity:
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we have:
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equivalent with:
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and this leads to:
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