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Question: Let \(S(K) = 1 + 3 + 5........ + (2K - 1) = 3 + {K^2}\). Then which of the following is true, A. P...

Let S(K)=1+3+5........+(2K−1)=3+K2S(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)⇒S(K+1)S(K) \Rightarrow S(K + 1)
C. S(K)≠S(K+1)S(K) \ne S(K + 1)
D. S(1)S(1)is correct.

Explanation

Solution

according to the question we have to check which option is correct if S(K)=1+3+5........+(2K−1)=3+K2S(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+K2S(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)=S(1) = 2(1)−1=3+(1)22\left( 1 \right) - 1 = 3 + {\left( 1 \right)^2}
⇒1≠4\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.
⇒S(K)=1+3+5........+(2K−1)=3+K2\Rightarrow S(K) = 1 + 3 + 5........ + (2K - 1) = 3 + {K^2}
Now, we have to add (2k+1)\left( {2k + 1} \right)to the both side of the given expression,
⇒S(K)=1+3+5........+(2K−1)+2k+1=3+K2+2k+1\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 K2+2k+1{K^2} + 2k + 1 in the expression obtained in the solution step 2 is the perfect square of (K+1)\left( {K + 1} \right). So we can see that expression in the form of (K+1)\left( {K + 1} \right) as mentioned below.
⇒S(K)=3+(K+1)2\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)S(K + 1) as mentioned below.
⇒S(K)=S(K+1)\Rightarrow S(K) = S(K + 1)
Hence, satisfy the given expression S(K)=1+3+5........+(2K−1)=3+K2S(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+K2S(K) = 1 + 3 + 5........ + (2K - 1) = 3 + {K^2} then S(K)=S(K+1)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 (2k+1)\left( {2k + 1} \right) in the solution step to make the R.H.S of the given expression in the perfect square of(k+1)\left( {k + 1} \right).