Six speeds

A drilling machine is to have 6 speed ranging from 50 to 750 revolution per minute. If the speed forms a geometric progression, determine their values.

Result

a1 =  50 rpm
a2 =  85.939 rpm
a3 =  147.71 rpm
a4 =  253.88 rpm
a5 =  436.362 rpm
a6 =  750.007 rpm

Solution:

$a_{ 1 } = 50 = 50 \ \text { rpm }$
$a_{ 6 } = 750 \ rpm \ \\ \ \\ a_{ 6 } = q^5 \ a_{ 1 } \ \\ \ \\ q = \sqrt[5]{ a_{ 6 }/a_{ 1 }} = \sqrt[5]{ 750/50} \doteq 1.7188 \ \\ \ \\ \ \\ a_{ 2 } = q \cdot \ a_{ 1 } = 1.7188 \cdot \ 50 \doteq 85.9386 = 85.939 \ \text { rpm }$
$a_{ 3 } = q \cdot \ a_{ 2 } = 1.7188 \cdot \ 85.9386 \doteq 147.7095 = 147.71 \ \text { rpm }$
$a_{ 4 } = q \cdot \ a_{ 3 } = 1.7188 \cdot \ 147.7095 \doteq 253.8798 = 253.88 \ \text { rpm }$
$a_{ 5 } = q \cdot \ a_{ 4 } = 1.7188 \cdot \ 253.8798 \doteq 436.3618 = 436.362 \ \text { rpm }$
$a_{ 6 } = q \cdot \ a_{ 5 } = 1.7188 \cdot \ 436.3618 \doteq 750.0068 = 750.007 \ \text { rpm }$

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