Prove by induction that 1+2+3+.....+n = n(n+1)÷2

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1) base case: n=1

for n=1, the statement take the next form: $1=1(1+1)/2$ . After simplification we get $1=1$ , which means the base case is true.

2) Induction step: under the assupmtion that statement is true for n=k, we will prove that it is true for n=k+1.

for n=k we get $1+2+3+...+k=k(k+1)/2$

for n=k+1 we get: $1+2+3+...+k+(k+1)= k(k+1)/2 + k+1= (k(k+1) + 2(k+1))/2 = (k+1)(k+2)/2$

The statement has been proven.