Question

Prove/disprove with mathematical induction that for any positive integer, n:

In text form:

1 + 2 + . . . + n = (n*(n+1))/2

Please provide actual answer instead of a link to an answer that is incorrect...

Answer 1

Given Equation

1 + 2 + . . . + n = (n*(n+1))/2     for any positive integer n

Let P(n) = 1 + 2 + . . . + n = (n*(n+1))/2     , n >0                     

Step # 1:

Show the equation is true for n = 1,n=2,..

Assume n=1 and put it in equation A

Left Side :

=1

Right Side:

            =1*(1+1) /2 = 2/2 =1

Here Left side equals to Right Side, hence the P (1) is true

Step # 2:

We now assume that P(k) is true, by considering n=k. Substitute k instead of n in P(n)

            1 + 2 + . . . + k= (k*(k+1))/2        . Consider this as Equation 1

Step # 3:

Now we have to prove P(n) is true for n=k+1.

We have to add k+1 in both the sides of equation 1

Left Side :

=1+2+3+