There are several different ways to take an average of numbers such as the Arithmetic Mean (AM) , Geometric Mean (GM) and the Quadratic Mean (QM) .
For the averages of two distinct positive real numbers :
Which of the above statements are true?
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From the https://brilliant.org/wiki/power-mean-qagh/ , the quadratic mean is the largest, followed by the arithmetic mean, followed by the geometric mean in a possible sequence of the three means. Also let the distinct numbers be a and b .
Statement 1:
Assume that we can form an arithmetic sequence such that the three means are consecutive terms and k is the difference between terms so that Quadratic Mean = k + Arithmetic Mean 2 a 2 + b 2 = k + 2 a + b 2 a 2 + b 2 = ( 2 2 k + a + b ) 2 2 ( a 2 + b 2 ) = 4 k 2 + 4 k a + 4 k b + a 2 + b 2 + 2 a b a 2 − 2 a b + b 2 = 4 k a + 4 k b + 4 k 2 ( a − b ) 2 = 4 k ( a + b + k ) Also, Arithmetic Mean = k + Geometric Mean 2 a + b = k + a b ( a + b − 2 k ) 2 = 4 a b a 2 + 2 a b + b 2 − 4 k a − 4 k b + 4 k 2 = 4 a b a 2 − 2 a b + b 2 = 4 k a + 4 k b − 4 k 2 ( a − b ) 2 = 4 k ( a + b − k )
Comparing the two blue lines, we see that it only works if k = 0 . This means that a = b , which can't happen as they are distinct. Therefore an arithmetic sequence cannot be formed.
Statement 2:
Assume that we can form a geometric sequence such that the three means are consecutive terms and k is the ratio between terms so that Quadratic Mean = k × Arithmetic Mean 2 a 2 + b 2 = 2 k ( a + b ) 2 ( a 2 + b 2 ) = k 2 ( a + b ) 2 k 2 = ( a + b ) 2 2 ( a 2 + b 2 ) Also, Arithmetic Mean = k × Geometric Mean 2 a + b = k a b ( a + b ) 2 = 4 k 2 a b k 2 = 4 a b ( a + b ) 2 Equating the two green statements, and cross-multiplying we get 8 a b ( a 2 + b 2 ) = ( a + b ) 4 Which simplifies to ( a − b ) 4 = 0 This also implies a = b , meaning that a geometric sequence cannot be formed either