Math CentralQuandaries & Queries


Question from Bhavya, a student:

Prove by induction that if Xi >= 0 for all i, then

(Summation Xi from 1 to n)^2 >= Summation Xi^2 from 1 to n


First I suggest that you look at Victoria's response in The Principle of Mathematical Induction.

For her step 1. check that your statement is true for i = 1.

Step 2. assume that your statement is true for some n = k ≥ 1. Thus for your statement that is assume that for some k ≥ 1


Step 3. is the inductive step. Based on the assumption in step 2. you are to prove that

inequality               (*).

To prove this let inequalitythen the left side is (xi + y)2. Expand this using the binomial theorem and use the assumption in part 2. to conclude that statement (*) is true.

Perform step 4.

I hope thus helps,

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