Q55 / 170 · Manipal (MET) 2018 · Mathematics

If $P(n): 1+3+5+\ldots+(2 n-1)=n^2$ is

  • Correct answer
    A
    true for all $n \in N$
  • B
    true for $n\gt1$
  • C
    true for no $n$
  • D
    None of these

Solution

Since, $P(1): 1=1$ (true) $\begin{aligned} & \text { Let } \quad P(k) \\ & \therefore 1+3+5+\ldots+(2 k-1)=k^2 \\ & \therefore P(k+1) \\ & \equiv 1+3+5+\ldots+(2 k-1)+2 k+1 \\ & \\ & \\ & =k^2+2 k+1=(k+1)^2 \end{aligned}$
So, it holds for all $n$.

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

More Probability Questions