Question

If P (n) is the statement "n² + n is even", and if P (r) is true, then P (r + 1) is true.

Answer 1

$$\begin{gathered} P\left(n\right):n^2+n\mathrm{is}\mathrm{even}. \\ \mathrm{Also}, \\ P\left(r\right)\mathrm{is}\mathrm{true}. \\ Thus,r^2+r\mathrm{is}\mathrm{even}. \\ \mathrm{To}\mathrm{prove}:P(r+1)\mathrm{is}\mathrm{true}. \\ Now, \\ P(r+1)=(r+1{)}^2+r+1 \\ =r^2+1+2r+r+1 \\ =r^2+3r+2 \\ =r^2+r+2r+2 \\ =P(r)+2(r+1\left) \\ P\right(r)\mathrm{is}\mathrm{even}. \\ Also,2(r+1)\mathrm{is}\mathrm{even},as\mathrm{it}\mathrm{is}\mathrm{a}\mathrm{multiple}\mathrm{of}2. \\ \mathrm{Therefore},P(r+1)\mathrm{is}\mathrm{even}and\mathrm{true}. \end{gathered}$$