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Let $\mathrm{T}(k)$ be the statement $1+3+5+\ldots+$ $(2 k-1)=k^{2}+10$
Which of the following is correct?
Options:
Which of the following is correct?
Solution:
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Verified Answer
The correct answer is:
$\mathrm{T}(k)$ is true $\Rightarrow \mathrm{T}(k+1)$ is true
When $\mathrm{k}=1, \mathrm{LHS}=1$ but $\mathrm{RHS}=1+10=11$
$\therefore \mathrm{T}(1)$ is not true Let $\mathrm{T}(k)$ is true. That is
$\begin{array}{l}
1+3+5+\ldots .+(2 k-1)=k^{2}+10 \\
\text { Now, } 1+3+5+\ldots . .+(2 k-1)+(2 k+1) \\
=k^{2}+10+2 k+1=(k+1)^{2}+10
\end{array}$
$\therefore \mathrm{T}(k+1)$ is true.
That is $\mathrm{T}(k)$ is true $\Rightarrow \mathrm{T}(k+1)$ is true.
But $\mathrm{T}(n)$ is not true for all $n \in \mathbf{N},$ as $\mathrm{T}(1)$ is not true.
$\therefore \mathrm{T}(1)$ is not true Let $\mathrm{T}(k)$ is true. That is
$\begin{array}{l}
1+3+5+\ldots .+(2 k-1)=k^{2}+10 \\
\text { Now, } 1+3+5+\ldots . .+(2 k-1)+(2 k+1) \\
=k^{2}+10+2 k+1=(k+1)^{2}+10
\end{array}$
$\therefore \mathrm{T}(k+1)$ is true.
That is $\mathrm{T}(k)$ is true $\Rightarrow \mathrm{T}(k+1)$ is true.
But $\mathrm{T}(n)$ is not true for all $n \in \mathbf{N},$ as $\mathrm{T}(1)$ is not true.
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