Q44 / 125 · BITSAT 2012 · Mathematics

Let $\mathrm{T}(k)$ be the statement $1+3+5+\ldots+$ $(2 k-1)=k^{2}+10$ Which of the following is correct?

  • A
    $\mathrm{T}(1)$ is true
  • Correct answer
    B
    $\mathrm{T}(k)$ is true $\Rightarrow \mathrm{T}(k+1)$ is true
  • C
    $\mathrm{T}(n)$ is true for all $n \in \mathbf{N}$
  • D
    All above are correct

Solution

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.

Hence, the correct answer is option: 'B'.

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