Using mathematical induction, prove that the sum of the first ๐ positive integers isequal to ๐(๐+๐)๐ .
Question
Using mathematical induction, prove that the sum of the first n positive integers is equal to .
Solution
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.
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