Question

Let $S(K) = 1 + 3 + 5........ + (2K - 1) = 3 + {K^2}$. Then which of the following is true,
A. Principle of mathematical induction can be used to prove the formula.
B. $S(K) \Rightarrow S(K + 1)$
C. $S(K) \ne S(K + 1)$
D. $S(1)$is correct.

Answer 1

Hint: according to the question we have to check which option is correct if $S(K) = 1 + 3 + 5........ + (2K - 1) = 3 + {K^2}$.
So, first of all we have to check by option method to satisfy the given expression $S(K) = 1 + 3 + 5........ + (2K - 1) = 3 + {K^2}$.
Hence, which option satisfies the given expression in the question that is the answer of the question.

Complete answer:
Step 1: First of all we have to check by option (D) that it satisfies the given expression or not.
So, $S(1) = $$2\left( 1 \right) - 1 = 3 + {\left( 1 \right)^2}$
$ \Rightarrow 1 \ne 4$
So, in the above solution L.H.S is not equal to R.S Hence, option (D) is wrong.
Step 2: Now, we have to check by option (B) that it satisfies the given expression or not.
$ \Rightarrow S(K) = 1 + 3 + 5........ + (2K - 1) = 3 + {K^2}$
Now, we have to add $\left( {2k + 1} \right)$to the both side of the given expression,
 $ \Rightarrow S(K) = 1 + 3 + 5........ + (2K - 1) + 2k + 1 = 3 + {K^2} + 2k + 1$
Step 3: Now, we have to see that the term ${K^2} + 2k + 1$ in the expression obtained in the solution step 2 is the perfect square of $\left( {K + 1} \right)$. So we can see that expression in the form of $\left( {K + 1} \right)$ as mentioned below.
$ \Rightarrow S(K) = 3 + {\left( {K + 1} \right)^2}$
Step 4: Now, we can see that the R.H.S of the expression obtained in the solution step 3 is in the form of $S(K + 1)$ as mentioned below.
$ \Rightarrow S(K) = S(K + 1)$
Hence, satisfy the given expression $S(K) = 1 + 3 + 5........ + (2K - 1) = 3 + {K^2}$
Final solution: Hence, the given expression if let $S(K) = 1 + 3 + 5........ + (2K - 1) = 3 + {K^2}$ then $S(K) = S(K + 1)$ satisfy the expression.

Hence, Option (B) is correct.

Note: It is necessary that we have to check by option method to satisfy the given expression in the question.
It is necessary to add $\left( {2k + 1} \right)$ in the solution step to make the R.H.S of the given expression in the perfect square of$\left( {k + 1} \right)$.