Question

P(n): 1 + 3 + 5 + 7 + 9 + .... + (2n - 1) = $n^2$ is true for which of the following statements?

• A. n $\in$ N
• B. n $\in$ N-{1}
• C. n $\in$ Even numbers
• D. n $\in$ N - {4}

Answer 1

The correct option is A n $\in$ N

$\mathrm{Consider}\mathrm{the}\mathrm{statement}P\left(n\right).P\left(n\right):1+3+5+...+(2n-1)=n^2\mathrm{When}n=1,\mathrm{LHS}=1\mathrm{and}\mathrm{RHS}=1^2=1\therefore P\left(1\right)\mathrm{is}\mathrm{true}.$
$\mathrm{Let}\mathrm{the}\mathrm{statement}\mathrm{be}\mathrm{true}\mathrm{for}\mathrm{some}\mathrm{positive}\mathrm{integer}k,i.e.,1+3+5+...+\left(2k-1\right)=k^2...\left(1\right)\mathrm{Then}\mathrm{we}\mathrm{have}\mathrm{to}\mathrm{prove}\mathrm{that}P\left(k+1\right)\mathrm{is}\mathrm{true}.\mathrm{Consider},\left(1+3+5+...+\left(2k-1\right)\right)+\left(2k+1\right)=k^2+2k+1\left(\mathrm{Using}\left(1\right)\right]={\left(k+1\right)}^2\mathrm{Thus},P\left(k+1\right)\mathrm{is}\mathrm{true}\mathrm{whenever}P\left(k\right)\mathrm{is}\mathrm{true}.\mathrm{Hence},\mathrm{by}\mathrm{principal}\mathrm{of}\mathrm{mathematical}\mathrm{induction},P\left(n\right)\mathrm{is}\mathrm{true}\mathrm{for}\mathrm{all}n\in N.$