9. (a) A coin has probability p of showing head when tossed. It is tossed n times. Let pn , denote the probability that no two (or more) consecutive heads occur. Prove that $\mathrm { p } 1 = 1 , \mathrm { p } ^ { 2 } = 1 - \mathrm { p } _ { 2 }$ and $\mathrm { p } = ( 1 -$ p ). $\mathrm { pn } - 1 + \mathrm { p } ( 1 - \mathrm { p } ) \mathrm { pn } - 2$ for all $\mathrm { n } \geq 3$.\\
(b) In (a), prove by induction on n , that $\mathrm { pn } = \mathrm { A } \alpha \mathrm { n } + \mathrm { B } \beta \mathrm { n }$ for all $\mathrm { n } \geq 1$, where $\alpha$ and $\beta$ are the roots of the quadratic x2-(1-p) $x - p ( 1 - p ) = 0$ and

$$A = \frac { p ^ { 2 } + \beta - 1 } { \alpha \beta - \alpha ^ { 2 } } , B = \frac { p ^ { 2 } + \alpha - 1 } { \alpha \beta - \beta ^ { 2 } } .$$

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  \item Let ABC and PQR be any two triangles in the same plane. Assume that the perpendiculars from the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ to the sides $\mathrm { QR } , \mathrm { RP } , \mathrm { PQ }$ respectively are concurrent. Using vector methods or otherwise, prove that the perpendiculars from $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ to $\mathrm { BC } , \mathrm { CA } , \mathrm { AB }$ respectively are also concurrent.
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