▁numbers , ▁such ▁as ▁ 2 ▁and ▁ 8 , ▁is ▁obtained ▁by ▁finding ▁a ▁value ▁A ▁such ▁that ▁ 2 ▁+ ▁ 8 ▁= ▁A ▁+ ▁A . ▁One ▁may ▁find ▁that ▁A ▁=   ( 2   +   8 )/ 2 ▁=   5 . ▁Switch ing ▁the ▁order ▁of ▁ 2 ▁and ▁ 8 ▁to ▁read ▁ 8 ▁and ▁ 2 ▁does ▁not ▁change ▁the ▁resulting ▁value ▁obtained ▁for ▁A . ▁The ▁mean ▁ 5 ▁is ▁not ▁less ▁than ▁the ▁minimum ▁ 2 ▁nor ▁greater ▁than ▁the ▁maximum   8 . ▁If ▁we ▁increase ▁the ▁number ▁of ▁terms ▁in ▁the ▁list ▁to ▁ 2 , ▁ 8 , ▁and ▁ 1 1 , ▁the ▁arithmetic ▁mean ▁is ▁found ▁by ▁solving ▁for ▁the ▁value ▁of ▁A ▁in ▁the ▁equation ▁ 2   +   8   +   1 1 ▁=   A   +   A   +   A . ▁One ▁finds ▁that ▁A ▁=   ( 2   +   8   +   1 1 )/ 3 ▁=   7 . ▁ ▁Ge ometric ▁mean ▁The ▁geometric ▁mean ▁of ▁n ▁positive ▁numbers ▁is ▁obtained ▁by ▁multip lying ▁them ▁all ▁together ▁and ▁then ▁taking ▁the ▁n th ▁root . ▁ ▁In ▁algebraic ▁terms , ▁the ▁geometric ▁mean ▁of ▁a 1 ,   a 2 ,   ...,   an ▁is ▁defined ▁as ▁▁▁▁ ▁Ge ometric ▁mean ▁can ▁be ▁thought ▁of ▁as ▁the ▁an til og ▁of ▁the ▁arithmetic ▁mean ▁of ▁the ▁logs ▁of ▁the ▁numbers . ▁