In a harmonic sequence, the first term is , and the third term is . The harmonic mean 1 2 1 8 between the two terms is

Answer 1

H = 2xy/(x+y)

x=1/2, y=1/8

Harmonic Mean = H = 2(1/2)(1/8)/(1/2+1/8) = (1/8)/(5/8) = 1/5

alternative way to calculate H:

Arithmetic Mean = the common simple average = A = sum divided by two = (1/2+1/8)/2 = 5/16

Geometric Mean = G = square root of the product = sqr[(1/2)(1/8)] = sqr(1/16) = 1/4

H = G^2/A = (1/16)/(5/16) = 1/5

for a harmonic sequence, each term is the harmonic mean of the two surrounding terms

1/2, 1/5, 1/8, ... would be 3 terms of a harmonic sequence, as 1/5 is the harmonic mean of 1/2 and 1/8

1/2, 1/5, 1/8, 1/11, 1/14, 1/17, 1/20, ...

the denominators increase by 3 each term

or if by chance, you meant 1/2, 1/8 as the first two terms then 1/8 would be the harmonic mean between 1/2 and the 3rd term. the 3rd term would be y where 1/8 =2(1/2)y/(1/2+y)

cross multiply

8y = 1/2 +y

7y = 1/2

y =1/14

then the harmonic sequence is 1/2, 1/8, 1/14, 1/20, 1/26, ...

the denominators increase by 6 each term