Question

Show by induction that 1 + 3 + 5 + · · · + (2n − 1) = n^2 for all positive integer n

Answer 1

Mathematical induction is a technique of proving a statement , theorem or formula which is thought to be true, for each and every natural number n.

Principle of Mathematical Induction Solution and Proof

Consider a statement P(n), where n is a natural number. Then to determine the validity of P(n) for every n, use the following principle:

Step 1: Check whether the given statement is true for n = 1.

Step 2: Assume that given statement P(n) is also true for n = k, where k is any positive integer.

Step 3: Prove that the result is true for P(k+1) for any positive integer k.

If the above-mentioned conditions are satisfied, then it can be concluded that P(n) is true for all n natural numbers.

Let P(n): 1 + 3 + 5 + ..... + (2n - 1) = n^2 be the given statement

Step 1: Check whether the given statement is true for n = 1

Put n = 1
Then, L.H.S = 1
R.H.S = (1)^2 = 1
∴. L.H.S = R.H.S.
⇒ P(n) is true for n = 1

Step 2: Assume that P(n) is true for n = k.
1 + 3 + 5 + … + (2k−1) = k²

Step 3: Check for n = k + 1

i.e. 1 + 3 + 5 + … + (2(k+1)−1) = (k+1)²

We can write the above equation as, ( before (2(k+1)−1) is (2k−1) )

1 + 3 + 5 + … + (2k−1) + (2(k+1)−1) = (k+1)²

Using step 2 result, we get (1 + 3 + 5 + … + (2k−1)= k² )

k² + (2(k+1)−1) = (k+1)²

k² + 2k + 2 −1 = (k+1)²

k² + 2k + 1 = (k+1)²

(k+1)² = (k+1)²

L.H.S. and R.H.S. are same.

So the result is true for n = k+1

∴ by the principle of mathematical induction P(n) is true for all natural numbers 'n'
Hence, 1 + 3 + 5 + ..... + (2n - 1) =n^2, for all n ϵ n

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