Given the series 1+2+3+4+5+....+n
Show that Sn=n(n+1) /2
This can be done using mathematical induction.
First, let's check the base case where n = 1.
When n = 1, the series becomes: 1 = 1(1+1)/2
Simplifying the right side, we get 1 = 1(2)/2, which is true.
Now, let's assume that the formula is true for some positive integer k, so that the sum of the first k natural numbers is given by Sk = k(k+1)/2.
We want to prove that the formula is also true for k+1, so the sum of the first (k+1) natural numbers should be given by (k+1)((k+1)+1)/2.
Now, we add (k+1) to both sides of the equation for Sk:
Sk + (k+1) = k(k+1)/2 + (k+1)
Combining the terms on the right side, we get:
Sk + (k+1) = [k(k+1) + 2(k+1)]/2
Factoring out (k+1) from the numerator, we have:
Sk + (k+1) = [(k+1)(k+2)]/2
Finally, simplifying the right side, we get:
Sk + (k+1) = (k+1)(k+2)/2
Since this is the same as (k+1)((k+1)+1)/2, we have shown that if the formula is true for k, then it is also true for k+1.
Therefore, by mathematical induction, the formula Sn = n(n+1)/2 holds for all positive integers n.