Consider the sequence of numbers $a_{0}, a_{1}, a_{2}, ...., a_{n}$ where $a_{0} = 17.23, a_{1} = 33.23$ and $a_{r + 2} = \frac{1}{2} (a_{r} + a_{r + 1})$ then

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$| a_{8} - a_{7} | = \frac{1}{8}$

$a_{0} - a_{1}, a_{1} - a_{2}, a_{2} - a_{3}, ......$ are in GP

$| a_{10} - a_{9} | = | a_{8} - a_{7} |$

$a_{0} - a_{2}, 2 (a_{1} - a_{2}), a_{1} - a_{3}, ......$ are in HP

answer is A, B, C.

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Detailed Solution

Let, $a_{r + 2} = \frac{1}{2} (a_{r} + a_{r + 1})$

$\Rightarrow 2 a_{r + 2} = a_{r} + a_{r + 1} \Rightarrow 2 (a_{r + 2} - a_{r + 1}) = a_{r} - a_{r + 1} \Rightarrow a_{r + 2} - a_{r + 1} = - \frac{1}{2} (a_{r + 1} - a_{r}) P u t, r = 8, 7, 6, 5, 4...., 3, 2, 1, 0 a_{10} - a_{9} = - \frac{1}{2} (a_{9} - a_{8}) = (- \frac{1}{2}) (- \frac{1}{2}) (a_{8} - a_{7}) = (- \frac{1}{2}) (- \frac{1}{2}) (- \frac{1}{2}) (a_{7} - a_{6})$