Question

Consider the following statement $P\left(n\right):1+3+5+...+\left(2n-1\right)=3+n^2,n\in N$
then which of the folowing is true?

• A. P(1) is correct.
• B. P(1) is not correct.
• C. P(n) $\Rightarrow$P(n+1)
• D. P(n) $\nRightarrow$P(n+1)

Answer 1

Answer: (B), (C)

P(n) $\Rightarrow$P(n+1)

$\mathrm{Consider}P\left(1\right)\mathrm{LHS}=1\mathrm{and}\mathrm{RHS}=3+1=4\mathrm{So},\mathrm{LHS}\ne \mathrm{RHS}\mathrm{for}n=1\mathrm{Given}P\left(n\right):1+3+5+...+\left(2n-1\right)=3+n^2\mathrm{Assume}P\left(k\right)\mathrm{is}\mathrm{true},i.e.,1+3+5+...+\left(2k-1\right)=3+k^2...\left(1\right)\mathrm{Consider}1+3+5+...+\left(2k-1\right)+\left(2\left(k+1\right)-1\right)=1+3+5+...+{\left(2k-1\right)}^2+\left(2k+1\right)=3+k^2+\left(2k+1\right)...\left(\mathrm{Using}\left(1\right)\right]=3+{\left(k+1\right)}^2\mathrm{Thus},P\left(k+1\right)\mathrm{is}\mathrm{true}\mathrm{when}P\left(k\right)\mathrm{is}\mathrm{true}.\mathrm{Although}P\left(1\right)\mathrm{is}\mathrm{not}\mathrm{true},\mathrm{assuming}P\left(k\right)\mathrm{to}\mathrm{be}\mathrm{true}\mathrm{implies}P\left(k+1\right)\mathrm{is}\mathrm{true}.$