JEE Main 2004 — Mathematical Induction Question with Solution

Given $$S(K)$$ $$=1+3+\mathrm{5...}+\left(2K-1\right)=3+K^2$$

When k = 1, S(1): 1 = 3 + 1,

L.H.S of S(k) $$\ne$$ R.H.S of S(k)

So S(1) is not true.

As S(1) is not true so principle of mathematical induction can not be used.

S(K+1) = 1 + 3 + 5... + (2K - 1) + (2K + 1) = 3 + (k + 1)²

Now let S(k) is true

$$\therefore$$ 1 + 3 + 5 +........(2k - 1) = 3 + k²

$$\Rightarrow$$ 1 + 3 + 5 +........(2k - 1) + (2k + 1) = 3 + k² + 2k +1

= 3 + (k + 1)²

$$\Rightarrow$$ S(k + 1) is true.

$$\therefore$$ S(k) $$\Rightarrow$$ S(k + 1)