Question
Let $$S\left( k \right) = 1 + 3 + 5 + ...... + \left( {2k - 1} \right) = 3 + {k^2}.$$ Then which of the following is true
A.
Principle of mathematical induction can be used to prove the formula
B.
$$S\left( k \right) \Rightarrow S\left( {k + 1} \right)$$
C.
$$S\left( k \right) ⇏ S\left( {k + 1} \right)$$
D.
$$S(1)$$ is correct
Answer :
$$S\left( k \right) \Rightarrow S\left( {k + 1} \right)$$
Solution :
$$\eqalign{
& S\left( k \right) = 1 + 3 + 5 + ...... + \left( {2k - 1} \right) = 3 + {k^2} \cr
& S\left( 1 \right):1 = 3 + 1,\,\,{\text{which is not true}} \cr
& \because \,\,S\left( 1 \right)\,\,{\text{is not true}}{\text{.}} \cr
& \therefore \,\,{\text{P}}{\text{.M}}{\text{.I}}{\text{. cannot be applied}} \cr
& {\text{Let }}S\left( k \right){\text{ is true, i}}{\text{.e}}{\text{.,}} \cr
& 1 + 3 + 5 + ...... + \left( {2k - 1} \right) = 3 + {k^2} \cr
& \Rightarrow \,\,1 + 3 + 5 + ...... + \left( {2k - 1} \right) + 2k + 1 \cr
& = 3 + {k^2} + 2k + 1 = 3 + {\left( {k + 1} \right)^2} \cr
& \therefore \,\,S\left( k \right) \Rightarrow S\left( {k + 1} \right) \cr} $$