The line `x+y=1` meets X-axis at A and Y-axis at B,P is the mid-point of `AB,P_(1)` is the foot of perpendicular from P to `OA, M_(1)` is that of `P_(1)` from `OP,P_(2)` is that of `M_(1)` from `OA,M_(2)` is that of `P_(2)` from `OP,P_(3)` is that of `M_(2)` from OA and so on. If `P_(n)` denotes theb nth foot of the perpendicular on OA, then find `OP_(n)`
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Explanation

We have, ` (OM_(n-1))^(2)=(OP_(n))^(2)+(P_(n)M_(n-1))^(2)`
` (OM_(n-1))^(2)=(OP_(n))^(2)+(OP_(n))^(2)=2(OP_(n))^(2)=2alpha _(n)^(2) [" say "`
Also, ` (OP_(n-1))^(2)=(OM_(n-1))^(2) +(P_(n-1)M_(n-1))^(2)`
`implies alpha _(n-1)^(2)= alpha _(n)^(2)+(1)/(2)alpha _(n-1)^(2) implies alpha _(n)^(2)=(1)/(4)alpha _(n-1)^(2)`
`implies alpha _(n)=(1)/(2)alpha _(n-1)`
`implies OP_(n)= alpha _(n)=(1)/(2^(2))alpha _(n-2)="..."=(1)/(2^(n))`
`therefore OP_(n)=((1)/(2))^(n)`