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If a(1)=a, a(2)=b, a(n+2)=(a(n+1)+a(n))/...

If `a_(1)=a, a_(2)=b, a_(n+2)=(a_(n+1)+a_(n))/2, nge 0` then `lim_(n to oo) a_(n)` is `(a+lamda_(1)b)/(lamda_(2)), (lamda_(1),lamda_(2) epsilonI)` where `lamda_(1)+lamda_(2)` is _______

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The correct Answer is:
5
`lim_(nto oo) a_(n)=a+(b-a)/2+(b-a)/8+…=a+2/3(b-a)=(a+2b)/3`
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The matrix A={:[(lamda_(1)^(2),lamda_(1)lamda_(2),lamda_(1)lamda_(3)),(lamda_(2)lamda_(1),lamda_(2)^(2),lamda_(2)lamda_(3)),(lamda_(3)lamda_(1),lamda_(3)lamda_(2),lamda_(3)^(2))]:} is idempotent if lamda_(1)^(2)+lamda_(2)^(2)+lamda_(3)^(2)=k where lamda_(1),lamda_(2),lamda_(3) are non-zero real numbers. Then the value of (10+k)^(2) is . . .

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