Let S(K) [ = 1 + 3 + 5...

Question

Let $S(K)$ $$ = 1 + 3 + 5... + \left( {2K - 1} \right) = 3 + {K^2}.$$  Then which of the following is true

Accepted Answer

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

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)

Solutions · 1