Prove that
Sn=u+a/t(2n-1)
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Proving that 1+2+3+...+n is n(n+1)/2
We give three proofs here that the n-th Triangular number, 1+2+3+...+n is n(n+1)/2. The first is a visual one involving only the formula for the area of a rectangle. This is followed by two proofs using algebra. The first uses "..." notation and the second introduces you to the Sigma notation which makes the proof more precise.
A visual proof that 1+2+3+...+n = n(n+1)/2
We can visualize the sum 1+2+3+...+n as a triangle of dots. Numbers which have such a pattern of dots are called Triangle (or triangular) numbers, written T(n), the sum of the integers from 1 to n :
n123456T(n) as a sum11+21+2+31+2+3+41..51..6T(n) as a triangle... T(n)=136101521
For the proof, we will count the number of dots in T(n) but, instead of summing the numbers 1, 2, 3, etc up to n we will find the total using only one multiplication and one division!
To do this, we will fit two copies of a triangle of dots together, one red and an upside-down copy in green.
E.g. T(4)=1+2+3+4
We give three proofs here that the n-th Triangular number, 1+2+3+...+n is n(n+1)/2. The first is a visual one involving only the formula for the area of a rectangle. This is followed by two proofs using algebra. The first uses "..." notation and the second introduces you to the Sigma notation which makes the proof more precise.
A visual proof that 1+2+3+...+n = n(n+1)/2
We can visualize the sum 1+2+3+...+n as a triangle of dots. Numbers which have such a pattern of dots are called Triangle (or triangular) numbers, written T(n), the sum of the integers from 1 to n :
n123456T(n) as a sum11+21+2+31+2+3+41..51..6T(n) as a triangle... T(n)=136101521
For the proof, we will count the number of dots in T(n) but, instead of summing the numbers 1, 2, 3, etc up to n we will find the total using only one multiplication and one division!
To do this, we will fit two copies of a triangle of dots together, one red and an upside-down copy in green.
E.g. T(4)=1+2+3+4
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