Let S(K):1+3+5+…+(2K−1)=3+k2Then which of the following is true?

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

1. A

$S\left(K\right)\ne S\left(K+1\right)$

2. B

$S\left(K\right)⇒S\left(K+1\right)$

3. C

$S\left(1\right)$ is correct

4. D

principle of mathematical induction can be used to prove the formula

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### Solution:

$S\left(1\right):1=3+{1}^{2}$ which is not true. Suppose

$S\left(K\right)$  is true, then

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

Adding  to both the sides, we get

$\begin{array}{l}1+3+5+\dots +\left(2K-1\right)+\left(2K+1\right)=3+{K}^{2}+2K+1\\ =3+\left(K+1{\right)}^{2}\end{array}$

which is

Thus,

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