How to Find Missing Numbers in Harmonic Progression

To find missing numbers in a Harmonic Progression (HP), convert the terms to their reciprocals, forming an Arithmetic Progression (AP). Identify the common difference in the AP, then use it to calculate the missing terms in the AP. Finally, take the reciprocal of these calculated terms to get the missing terms in the original HP sequence.

Let's discuss this in detail.

Harmonic Progression Definition

The Harmonic Progression (HP) is a sequence of terms in which the reciprocals form an arithmetic progression (AP).

If a₁, a₂, a₃, . . . are terms of the harmonic progression then their reciprocals \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \ldots will form an arithmetic progression.

Some examples include:

• 1, 1/2, 1/3, 1/4, 1/5, . . .
  • The reciprocals (1, 2, 3, 4, 5, ...) form an Arithmetic Progression (AP).
• 6, 3, 2, 1.5, . . .
  • The reciprocals (1/6, 1/3, 1/2, 2/3, ...) form an AP.
• 20, 15, 12, 10, . . .
  • The reciprocals (1/20, 1/15, 1/12, 1/10, ...) form an AP.

Harmonic Progression Formula

The general term of a harmonic progression can be calculated by the first finding the general term of its corresponding arithmetic progression. If \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \ldots forms an AP with the common difference d the general term of the corresponding HP is given by:

T_n = \frac{1}{a_1 + (n - 1) \cdot d}

Where,

• T_n is the n^th term of the HP,
• a₁ is the first term of the corresponding AP,
• d is the common difference of the AP, and
• n is the number of terms.

How to Find Missing Number in Harmonic Progression?

To find missing numbers in a Harmonic Progression (HP), follow these steps:

1. Convert HP to AP: Take the reciprocals of the given HP terms to form an Arithmetic Progression (AP).
2. Identify the Common Difference: Find the common difference in the AP sequence (subtract consecutive terms).
3. Calculate the Missing Term: Use the AP formula a_n = a₁ + (n − 1)d to find the missing terms in the AP.
4. Convert Back to HP: Take the reciprocals of the missing AP terms to get the missing HP terms.

Solved Examples

Example 1: Find the missing term in the sequence: 6, _, 3.

Solution:

The given HP sequence is 6, _, 3.

Convert the terms to their reciprocals to form an AP: 1/6, _, 1/3

Difference between first and third term of an AP is 2d (a, a + d, a + 2d, . . .)

Thus, 2d = 1/3 − 1/6 = 2/6 − 1/6 = 1/6

⇒ d = 1/12

Let the missing term be x. In the AP:

x = 1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4

So, the missing term in AP is 1/4​.

Take the reciprocal of 1/4 to get the missing HP term:

x = 4

Thus, the missing term in the HP is 4. The complete sequence is: 6, 4, 3 (HP)

Example 2: Find the missing term in the sequence: 1/10, _, 1/5.

Solution:

The given HP sequence is 1/10, _, 1/5.

Convert the terms to their reciprocals to form an AP: 10, _, 5

Difference between first and third term of an AP is 2d (a, a + d, a + 2d, . . .)

Thus, 2d= 5 − 10 = − 5

⇒ d = -2.5

Let the missing term be x. In the AP:

x = 10 - 2.5 = 7.5

The missing term in AP is 7.5​.

Reciprocal of 7.5​ is 7.5.

Final HP sequence: 10, 7.5, 5.

Conclusion

Finding missing numbers is a fundamental skill in the mathematics applicable in the various scenarios from the simple arithmetic to the complex problem-solving. Understanding the methods and practicing with the different types of problems can help strengthen this skill enabling the students and professionals alike to the tackle mathematical challenges effectively.

Read More,

• Harmonic Progression
• Arithmetic Progression
• Geometric Progression