What is a Harmonic Progression?
A Harmonic Progression (HP) is a sequence of numbers whose reciprocals form an Arithmetic Progression (AP).
The Inverted AP Model
Consider the sequence: 1, 1/2, 1/3, 1/4, 1/5... Take the reciprocal of each term to get: 1, 2, 3, 4, 5... which is a standard AP with $d = 1$.
Although Harmonic Progression is tested less frequently than AP and GP, it completes the study of classical progressions and appears in average speed problems (equal distances), physics, acoustics, and mixed Modern Math exam questions.
Connected Vault Modules:
The harmonic mean of two numbers $a$ and $b$. Exact formula for equal-distance average speed.
A sequence is an HP if and only if the difference of consecutive reciprocals is constant.
Verifying an HP Sequence
Q: Is the sequence 1, 1/2, 1/3, 1/4, 1/5, ... an HP?
Step 1: Take reciprocals of each term: 1, 2, 3, 4, 5, ...
Step 2: Check common difference: $2 - 1 = 1, 3 - 2 = 1, 4 - 3 = 1$.
Conclusion: Since reciprocals form an AP with $d = 1$, the original sequence is an HP (Yes).
Finding the Next Term of an HP
Q: Find the next term of the sequence: 1/3, 1/5, 1/7, ...
Step 1: Take reciprocals to get AP: 3, 5, 7, ...
Step 2: AP first term $a = 3$, common difference $d = 2$.
Step 3: Next AP term (4th term) $= 7 + 2 = 9$.
Step 4: Invert back to get 4th HP term: $$\mathbf{\frac{1}{9}}$$
Finding the 10th Term of an HP
Q: The reciprocals of an HP have a first term of 4 and a common difference of 2. Find the 10th term of the HP.
Step 1: Reciprocals form an AP with $a = 4$ and $d = 2$.
Step 2: Calculate 10th AP term: $a_{10} = a + (10 - 1)d = 4 + 9 \times 2 = 4 + 18 = 22$.
Step 3: Take the reciprocal to get 10th HP term: $$H_{10} = \mathbf{\frac{1}{22}}$$
❌ Trap 1: Subtracting HP Terms
H_2 - H_1 ≠ d
HP terms DO NOT have a constant difference. Always take reciprocals ($1/H_2 - 1/H_1$) before subtracting!
❌ Trap 2: Direct AP Sum on HP
Sum of HP ≠ 1 / Sum of AP
There is no simple direct formula for the sum of an HP series. You cannot just sum the AP and invert it!
❌ Trap 3: Confusing HP with GP
Fractions ≠ Automatic HP
Having fractions does not imply an HP. Check if denominators form an AP (constant difference) or GP (constant ratio)!
⚡ The Flip-Solve-Flip Technique
Never memorize separate HP formulas. Take reciprocals, use simple AP rules, then invert the final term!
⚡ Harmonic Mean Average Speed Shortcut
When covering equal distances at speeds $a$ and $b$, average speed is the Harmonic Mean of the two speeds, NOT the arithmetic mean!
Question 1:
Determine whether the sequence 1/2, 1/4, 1/6, 1/8, ... forms an HP.
Solution: Take reciprocals: 2, 4, 6, 8, ...
Differences: 4 - 2 = 2, 6 - 4 = 2, 8 - 6 = 2. Reciprocals form an AP ⇒ Yes, it is an HP.
Question 2:
Find the next term of the HP sequence: 1/5, 1/8, 1/11, ...
Solution: Reciprocals form AP: 5, 8, 11, ... ($a=5, d=3$).
4th AP term = 11 + 3 = 14 ⇒ 4th HP term = 1/14.
Question 3:
If the reciprocals of an HP form the AP 6, 10, 14, ..., find the 7th term of the HP.
Solution: AP first term $a = 6$, common difference $d = 4$.
7th AP term = 6 + 6(4) = 6 + 24 = 30 ⇒ 7th HP term = 1/30.
Q: Is every sequence of fractions an HP?
No. A sequence is an HP only if its reciprocals form an Arithmetic Progression (constant difference).
Q: Do I need to memorize a separate set of HP formulas?
No. Most HP problems are solved by converting the reciprocals into an AP, solving using standard AP formulas, and taking the reciprocal of the final answer.
Q: What is the main practical application of HP in CAT?
Harmonic Mean ($HM = 2ab / (a+b)$) is used for calculating the average speed of a journey when equal distances are traveled at different speeds.