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Correct option is A)

$Mu=Mv_{1}+2Mv_{2}β$_____ (i)

and $21βMu_{2}=21βMv_{1}+21β2Mβv_{2}$_____ (ii)

Solving (i) and (ii)

$v_{2}=34βu$

$v_{2}=34βu$

Similarly for second and third ball

$2MβΓ34βu=2Mβv_{2}+2_{2}Mβv_{3}$

$2MβΓ34βu=2Mβv_{2}+2_{2}Mβv_{3}$

and $21β2Mβ(34βu)_{2}$$=21β2Mβ(v_{2})_{2}+21β(2)_{2}Mβ(v_{3})_{2}$

Solving $v_{3}=(a_{r}β)u$

in this way, $v_{n}=(34β)_{nβ1}u$

For nth ball to loop $v_{n}=5gRβ$

i.e.,$(34β)_{nβ1}u=5gRβ$

or $u=(43β)_{nβ1}5gRβ$

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