Question

Let we have the sum of a series as \[S\left( K \right)=1+3+5...+\left( 2K-1 \right)=3+{{K}^{2}}\]. Then which of the following is true
(A). Principle of mathematical induction can be used to prove the formula
(B). $S\left( K \right)=S\left( K+1 \right)$
(C). $S\left( K \right)\ne S\left( K+1 \right)$
(D). $S\left( 1 \right)$ is correct

Answer 1

Hint: For solving this question, we check all the options individually. For option (a), we use the series expansion to establish simplified form and check the equality. For option (b), simply obtain S (K + 1) by using the expansion. For option (c), if option (b) is false then option (c) is correct. For option (d), simply put K = 1. By using this methodology, we solve the problem.

Complete step-by-step solution -
As given in the problem statement, \[S\left( K \right)=1+3+5...+\left( 2K-1 \right)=3+{{K}^{2}}\].
Now, considering the series 1 + 3 + 5 + …... + (2k – 1), we can obtain a simplified expression by using the sum formula as the series is in A.P.
The sum formula of a series in A.P. is $\dfrac{n}{2}\left[ 2a+\left( n-1 \right)\cdot d \right]$. Now, replacing n = K, a = 1 and d = 2, we get
$\begin{align}
  & S\left( K \right)=\dfrac{K}{2}\left[ 2\times 1+\left( K-1 \right)\times 2 \right] \\
 & S\left( K \right)=\dfrac{K}{2}\left( 2+2k-2 \right) \\
 & S\left( K \right)=\dfrac{K}{2}\times 2K \\
 & S\left( K \right)={{K}^{2}} \\
\end{align}$
The obtained value is different from the given value ${{K}^{2}}\ne 3+{{K}^{2}}$, so option (a) is incorrect.
For option (b), $S\left( K \right)=3+{{K}^{2}}$, so replacing K = K + 1, we get
$\begin{align}
  & S\left( K+1 \right)=3+{{\left( K+1 \right)}^{2}} \\
 & S\left( K+1 \right)=3+{{K}^{2}}+2K+1 \\
 & S\left( K+1 \right)={{K}^{2}}+2K+4 \\
 & \therefore S\left( K+1 \right)\ne S\left( K \right) \\
\end{align}$
Hence, option (b) is incorrect.
For option (c), by using the result of option (b) we conclude that option (c) is correct.
For option (d), putting the value of K =1, we get
S (1) = 1 = 3 +1
S (1) = $1=4$, which is not true.
Therefore, S (1) is not true. Hence, option (d) is incorrect.
Therefore, option (c) is correct.

Note: Students must follow the stepwise procedure for solving this problem. This procedure ensures that no correct option is missed and hence reduces the chances of error. Problems related to mathematical induction are solved like this.