Using mathematical induction, prove that the sum of the first 𝒏 positive integers isequal to 𝒏(𝒏+𝟏)𝟐 .

Sure, here is the proof using mathematical induction:

Step 1: Base Case First, we need to prove the statement is true for n=1. The sum of the first 1 positive integer is 1, and 1*(1+1)/2 = 1. So, the statement is true for n=1.

Step 2: Inductive Step Next, we assume the statement is true for some arbitrary positive integer k. That is, we assume that the sum of the first k positive integers is k*(k+1)/2.

Step 3: Inductive Hypothesis We need to prove that the statement is true for n=k+1. The sum of the first k+1 positive integers is (k+1) + k*(k+1)/2 = (k+1)*(k+2)/2.

Step 4: Conclusion Since the statement is true for n=1 and if it's true for n=k then it's true for n=k+1, by the principle of mathematical induction, the statement is true for all positive integers n.