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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
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B
S(k) S(k+1)
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C
S(k) S(k+1)
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D
S(1) is correct
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Solution

The correct option is C $$S (k)$$ $$\Rightarrow$$ $$S (k + 1)$$
Putting $$k=1$$, we get L.H.S=1 and R.H.S=4. Hence $$A$$ and $$D$$ are incorrect.
Now, $$S\left( k+1 \right) =1+3+5+$$...$$+\left( 2k-1 \right) +\left( 2\left( k+1 \right) -1 \right) $$
    $$\Rightarrow S\left( k+1 \right) =S\left( k \right) +\left( 2k+1 \right) $$                                               [Since  $$S\left( k \right) =1+3+5+$$...$$+\left( 2k-1 \right)$$]
    $$\Rightarrow S\left( k+1 \right) =3+{ k }^{ 2 }+\left( 2k+1 \right) $$
    $$\Rightarrow S\left( k+1 \right) =3+{ \left( k+1 \right)  }^{ 2 }$$ 
 $$B$$ is correct option.

Mathematics

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