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.

Correct result:

a₁ =  50 rpm
a₂ =  85.9386 rpm
a₃ =  147.7088 rpm
a₄ =  253.8778 rpm
a₅ =  436.3581 rpm
a₆ =  750 rpm

Solution:

$a_1=50=50\text{ rpm}$

$$\begin{gathered} a_6=750\text{ rpm } \\ {a}_6=q^5{a}_1 \\ q=\sqrt[5]{a_6\mathrm{/}{a}_1}=\sqrt[5]{750\mathrm{/}50}\doteq 1.7188 \\ {a}_2=q\cdot a_1=1.7188\cdot 50=85.9386\text{ rpm} \end{gathered}$$

$a_3=q\cdot a_2=1.7188\cdot 85.9386=147.7088\text{ rpm}$

$a_4=q\cdot a_3=1.7188\cdot 147.7088=253.8778\text{ rpm}$

$a_5=q\cdot a_4=1.7188\cdot 253.8778=436.3581\text{ rpm}$

$a_6=q\cdot a_5=1.7188\cdot 436.3581=750\text{ rpm}$

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