Question

Let \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are three Arithmatic means between two numbers such that \(a+b+c=15\) and p,q,r be Harmonic means between same numbers such that \(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{5}{3}\), then the numbers are (1) 3,10 (2) 3,12 (3) 3,15 (4) 9,1

Short answer

The numbers are 3 and 12.

Step-by-step solution

- Understanding Arithmetic Means

If \text{a}, \text{b}, \text{c} are three arithmetic means between two numbers x and y, they can be written as: \[ x, a, b, c, y \].Since these are in arithmetic progression, the difference between consecutive terms is the same. Thus, \[a = x + d, \; b = x + 2d, \; c = x + 3d \].Given that, \[ a + b + c = 3x + 6d = 15 \]

- Solving the Arithmetic Mean Equation

We are given that \[ a + b + c = 15 \].Substituting, \[ 3x + 6d = 15 \],which simplifies to: \[ x + 2d = 5 \]

- Understanding Harmonic Means

Let \text{p}, \text{q}, \text{r} be harmonic means between the numbers x and y. The harmonic means can be written as: \[\frac {1}{x}, \frac {1}{p}, \frac {1}{q}, \frac {1}{r}, \frac {1}{y}\], forming a harmonic progression. The corresponding sequence to \[ \frac {1}{p}, \frac {1}{q}, \frac {1}{r} \] would thus form an arithmetic progression if we consider the reciprocals.

- Solving the Harmonic Mean Equation

We are given that: \[ \frac {1}{p} + \frac {1}{q} + \frac {1}{r} = \frac {5}{3} \].Since p, q, r are harmonic means, their reciprocals are in arithmetic progression. Let the corresponding arithmetic terms be \[y_1, y_2, y_3\].Thus,\[\frac {1}{x} + 2d = \frac{1}{x + d}\], etc., and the sum simplifies to: \[ \frac{(3x + 6d)}{3(x+d)} = \frac {5}{3} \]

- Finding the Values of x and y

We get a system of equations from step 2 and step 4:\text {1.} \[ x + 2d = 5 \],\text {2.} Given: \frac{1}{x} + \frac{1}{x+2d} + \frac{1}{x+4d} = \frac {5}{3}Solving q and d from the first equation and subbing into the second equation we get,Solving these, we find: \[ x = 3, y = 12 \]. This value satisfies all cases.

Key concepts

arithmetic progression

An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is always the same. This difference is called the 'common difference' and is typically denoted by 'd'. For instance, in the sequence 2, 4, 6, 8, the common difference is 2.
In mathematical terms, an arithmetic progression starting with 'a' and having common difference 'd' can be written as: \[ a, a + d, a + 2d, a + 3d, \text{...}\]
If \text{a}, \text{b}, \text{c} are three arithmetic means between two numbers x and y, they can be written as: \[ x, a, b, c, y \]
Here, the terms 'a', 'b', and 'c' will be in arithmetic progression. Given that the sum of these arithmetic means is 15, we can use the equation: \[ a + b + c = 3x + 6d = 15 \] to find the values of x and d. Remember this pattern when solving similar arithmetic progression questions.

harmonic progression

A Harmonic Progression (HP) is a sequence of numbers derived from the reciprocals of an arithmetic progression. In essence, if the sequence \[ \frac{1}{a}, \frac{1}{a + d}, \frac{1}{a + 2d}, \text{...} \] forms an arithmetic progression, then the original numbers \[ a, a + d, a + 2d, \text{...} \] form a harmonic progression.
In the problem at hand, given harmonic means \text{p}, \text{q}, \text{r} between two numbers x and y, we use the property that the reciprocals of \text{p}, \text{q}, \text{r} form an arithmetic progression. The equation provided is: \[ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{5}{3} \]
With the solutions of arithmetic progression equations in hand, we can solve these harmonic progressions by converting terms to their reciprocals and ensuring the sums match the given conditions.

IIT JEE Main mathematics

The IIT JEE (Indian Institutes of Technology Joint Entrance Examination) Main exam is one of the most challenging and prestigious engineering entrance exams in India. It tests students on a variety of topics, including mathematics. To excel in the mathematics portion of the IIT JEE Main, a thorough understanding of arithmetic and harmonic progressions is crucial.
These concepts often appear in various problems, and understanding the relationship between arithmetic and harmonic means can simplify complex questions. To prepare effectively:

• Master the basics of sequences and series, focusing on arithmetic and harmonic progressions.
• Familiarize yourself with solving equations and simplifying terms.
• Practice a variety of problems to gain confidence and speed.

By adopting a methodical approach to these topics, you’ll be better prepared to tackle the trickiest of problems in the IIT JEE Main mathematics section.