1/3 and 1/5
a₁=3 and a₇=5 (reciprocal; since the reciprocal of a harmonic sequence form a arithmetic sequence.)
Find d:
an= a₁ + (n-1) d
5= 3 + (7-1) d
5= 3 + (6)d
5 - 3= 6d
2 = 6d
6
1/3 = d
Find a₂: Find a₃: Find a₄:
an= a₁ + (n-1) d a₃= 3 + (3-1) 1/3 a₄= 3 + (4-1) 1/3
a₂= 3 + (2-1) 1/3 = 3 + (2) 1/3 = 3+ (3) 1/3
= 3 + (1) 1/3 = 3+ 2/3 = 3 + 1
= 3 + 1/3 = 11/ 3 = 4
= 10/3
Find a₅: Find a₆:
a₅= 3 + (5-1) 1/3 a₆= 3 + (6-1) 1/3
= 3 + (4) 1/3 = 3 + (5) 1/3
= 3 + 4/3 = 3 + 5/3
= 13/3 = 14/3
∴ the five harmonic means are 3/10, 3/11, 1/4, 3/13, and 3/14.
