One geometric mean a is inserted between the numbers 2 and 1024. Two geometric means b and c are inserted between 2 and a , a and 1024 respectively. Now four geometric means d, e, f, g are inserted between 2 and b, b and a, a and c, c and 1024 respectively and this process is continued till O is inserted. The value of $c^{4}$ is $2^{z}$ . The value of z is ___

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Solution

The given sequence will be:
2 h d i b j e k a l f m c n g o 1024

The question indirectly indicating that 26 GM's are inserted between 2 and 1024.
$2 G_{1} G_{2} G_{3} G_{4} G_{5} G_{6} G_{7} . . . . . . G_{15} 1024$
$T_{1} = 2$
$T_{17} = 1024$
Let the common ratio be r.
$2 \times r^{17 - 1} = 1024$
$r^{16} = 512$
$r = 2^{9 / 16}$
$T h e v a l u e o f c = a r^{12} = 2^{31 / 4}$
$c^{4} = 2^{31}$
$z = 31$

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