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Q)

$ If\;a_{1}=\frac{1}{2}\;,a_{k+1}=a_{k}^{2}+a_{k}\;\forall\;k\;\geq\;1\;and\;x_{n}=\large\frac{1}{a_{1}+1}+\large\frac{1}{a_{2}+1}+...\;\large\frac{1}{a_{n}+1}\;the \;value\;of\;[x_{50}]\;is\;([.]\;represents\;greatest\;integer\;function)$)

$(a)\;1\qquad(b)\;2\qquad(c)\;3\qquad(d)\;None\;of\;these$

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A)
Answer : (a) 1
Explanation : $\;a_{k+1}=a_{k}\;(a_{k}+1)$
$\large\frac{1}{a_{k+1}}=\large\frac{1}{a_{k}}-\large\frac{1}{a_{k+1}}$
$x_{n}=\large\frac{1}{a_{1}}-\large\frac{1}{a_{n+1}}=2-\large\frac{1}{a_{n+1}}$
$x_{50}=2-\large\frac{1}{a_{51}}$
$a_{51}\;\geq\;1$
$x_{50}=1\;.$
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