Practice Questions
Test your grasp of reciprocals and averages with these harmonic progression questions.
Find the 5th term of the harmonic progression 1/3, 1/7, 1/11, 1/15, ...
The reciprocals 3, 7, 11, 15 are in AP with $a=3$ and $d=4$. The 5th term of AP = $3+(4 \times 4)=19$. So, the 5th term of HP = 1/19.
What is the Harmonic Mean (HM) of 4 and 6?
HM = $(2 \times 4 \times 6) / (4+6) = 48/10 = 4.8$.
If a, b, c are in HP, then which of the following is true?
By definition, the middle term of an HP is the Harmonic Mean of the outer two terms.
An airplane flies a equilateral triangle path with speeds 200, 300, and 600 km/h on each leg. Find the average speed.
When distances are equal, average speed is the Harmonic Mean. HM = $3 / (1/200 + 1/300 + 1/600) = 3 / ((3+2+1)/600) = 3 / (6/600) = 3 \times 100 = 300$ km/h.
If 1/3, 1/5, 1/7… form an HP, find its 6th term.
Corresponding AP: 3, 5, 7… with $a=3, d=2$. 6th term of AP = $3 + 5(2) = 13$. 6th term of HP = $1/13$.
Insert two harmonic means between 2 and 6.
Corresponding AP: $1/2, 1/H_1, 1/H_2, 1/6$. $d = -1/9$. $1/H_1 = 1/2 - 1/9 = 7/18$, $1/H_2 = 7/18 - 1/9 = 5/18$. So $H_1 = 18/7$, $H_2 = 18/5$.
If a, b, c are in HP, then which relation holds?
For HP, the reciprocals are in AP: $2/b = 1/a + 1/c$.
The reciprocals of 4, 6, 12 form an AP. Are they in HP?
Reciprocals: $1/4, 1/6, 1/12$. Check: $2(1/6) = 1/3$ and $1/4 + 1/12 = 1/3$. They are in AP, so 4, 6, 12 are in HP.
Find the sum of first n terms of HP corresponding to AP 2, 4, 6…
HP terms are $1/2, 1/4, 1/6, … = 1/(2n)$. Sum = $1/2(1 + 1/2 + 1/3 + … + 1/n)$. This is the harmonic series with no closed form.
In an HP, if first term is 1/2 and common difference of corresponding AP is 3, find the 5th term.
Corresponding AP: $a=2, d=3$. 5th term of AP = $2 + 4(3) = 14$. So 5th term of HP = $1/14$.
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