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CAT To find the nth term of a HP - Practice Questions & MCQ

Edited By admin | Updated on Oct 04, 2023 04:20 PM | #CAT

Syllabus

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4 Questions around this concept.

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MEDIUM

Find the number of terms in the sequence $1,-1,\frac{-1}{3},\frac{-1}{5},\frac{-1}{7},......\frac{-1}{141}$

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Concepts Covered - 1

To find the nth term of a HP

Definition: A Harmonic Progression (HP) is a sequence of numbers in which the reciprocals of the numbers form an Arithmetic Progression (AP).

Given an HP, if you take the reciprocals of its terms, the resulting sequence will be an AP.

Formula to find the nth term of an HP:

If the given HP is: $\mathrm{\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},...}$

Then the corresponding AP is: $\mathrm{{a_1},{a_2},{a_3},...}$

The nth term of the HP can be found by taking the reciprocal of the nth term of the corresponding AP.

Foundation Building Questions:

Question 1: The first three terms of an HP are 1, 1/2, 1/3. Find the 4th term.

Solution:

The corresponding AP of the given HP is: 1, 2, 3

Using the formula for the nth term of an AP: $\mathrm{a_n=a_1+(n-1)\times d}$

For the 4th term, $\mathrm{a_4=1+(4-1)\times 1=4}$

The 4th term of the HP is the reciprocal of this: $\mathrm{\frac{1}{4}}$

Question 2: If the 2nd term of an HP is 1/3 and the 5th term is 1/6, find the 7th term.

Solution:

The corresponding AP for the given terms is: 3 (2nd term) and 6 (5th term).

Using the formula for the nth term of an AP:

Difference, $\mathrm{d = (6 - 3) / (5 - 2) = 1}$

For the 7th term, $\mathrm{a_7 = 3 + 5 \times 1 = 8}$

The 7th term of the HP is the reciprocal of this: $\mathrm{\frac{1}{8}}$

Application of Previous Concepts:

The main concept to apply when dealing with HPs is that of an AP. Since the reciprocals of the terms of an HP form an AP, understanding the properties and formulas related to APs (from Concepts 2 & 3) is crucial for solving problems related to HPs.

Tips and Tricks:

1. Reciprocal is Key: When working with HP, always remember to switch to its corresponding AP by taking the reciprocals of the terms.

2. Use AP Concepts: Since the core of solving HP problems lies in AP, be proficient in the properties, formulas, and techniques associated with APs.

3. Visual Representation: Plotting the terms of an HP can give a visual idea of its nature, especially if it’s converging or diverging.

4. Beware of Common Errors: Ensure you're always working with the correct sequence. It’s easy to forget and directly apply AP concepts to HP without considering the reciprocal relationship.

5. Real-world Application: HPs can represent real-world scenarios where values decrease in a specific manner, such as some depreciation methods.

6. Practice with Mixed Sequences: Sometimes, you might encounter problems that mix terms from AP, GP, and HP. Practise such problems to increase familiarity.

To successfully tackle problems related to Harmonic Progressions, it's vital to understand the intrinsic relationship between HPs and APs. By mastering the art of switching between these two sequences and leveraging knowledge from earlier concepts, one can approach HP-related problems with confidence.