Define a sequence `(a_(n))` by `a_(1)=5,a_(n)=a_(1)a_(2)…a_(n-1)+4` for `ngt1`. Then `lim_(nrarroo) (sqrt(a_(n)))/(a_(n-1))`
Define a sequence `(a_(n))` by `a_(1)=5,a_(n)=a_(1)a_(2)…a_(n-1)+4` for `ngt1`. Then `lim_(nrarroo) (sqrt(a_(n)))/(a_(n-1))`
A. Equals `1//2`
B. equals 1
C. equals 2/5
D. does not exist
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Correct Answer - C
`a_(1)=5`
`a_(n)=a_(1)a_(2)….a_(n-1)+4`
`a_(2)=a_(1)+4=9`
`a_(3)=a_(1).a_(2)+4=5xx9+4=49`
`a_(4)=a_(1)a_(2)a_(3)+4=2209`
`a_(5)=a_(1)a_(2)a_(3)a_(4)+4=4870849=(2207)^(2)`
`a_(5)=(a_(4)-2)^(2)`
`a_(4)=(49-2)^(2)=(a_(3)-2)^(2)`
`a_(3)=(9-2)^(2)=(a_(2)-2)^(2)`
`an=(a_(n-1)-2)^(2)`
`sqrt(a_(n))=a_(n-1)-2`
`(sqrt(a_(n)))/(a_(n-1))=(a_(n-1)-2)/(a_(n-1))=1-underset(nrarroo)"lim"(2)/(a_(n-1))`
`because underset(nrarroo)"lim"a_(n-1)=oo`
= 1
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