Question

Using PMI prove that 1+3+5.........+2n-1=n^2

Answer 1

It contains 2 steps.

Step 1:
prove that the equation is valid when n = 1

When n = 1, we have
( 2×1 - 1) = 1²,
so the statement holds for n = 1.

Step 2:
Assume that the equation is true for n, and prove that the equation is true for n + 1.

Assume:

1 + 3 + 5 + ... + (2n - 1)
= n^{2 -----(i)}


Prove:

1 + 3 + 5 +...+ (2(n + 1) - 1 = (n + 1)²

Proof:

1 + 3 + 5 +... + (2(n + 1) - 1)

= 1 + 3 + 5 + ... + (2n - 1) + (2n + 2 - 1) ----(ii)

but from (i)

1 + 3 + 5 + .. + (2n - 1) = n²

Substituting (i) in (ii)

gives,

1 + 3 + 5 + ... + (2n - 1) + (2n + 2 - 1)
=n² + (2n + 2 - 1)
= n² + 2n + 1
= (n + 1)<sup>2

So the statement holds for (n+1) also.</sup>

So, by induction, for every positive integer n,

1 + 3 + 5 + . + (2n - 1) = n².