Question

Find the harmonic mean of 5 and 3 . The following are the steps involved in solving the above problem. Arrange them in sequential order. (a) \(\mathrm{HM}=\frac{2 \times 5 \times 3}{5+3}\) (b) We know that the harmonic mean of \(\mathrm{a}, \mathrm{b}\) is \(\frac{2 \mathrm{ab}}{\mathrm{a}+\mathrm{b}}\) (c) Here \(\mathrm{a}=5\) and \(\mathrm{b}=3\) (d) \(\mathrm{HM}=\frac{30}{8}=\frac{15}{4}\) (1) bcda (2) bcad (3) abcd (4) badc

Short answer

Question: Based on the aforementioned step-by-step solution, find the harmonic mean of 5 and 3. Answer: The harmonic mean of 5 and 3 is $$\frac{15}{4}$$, or 3.75.

Step-by-step solution

Select the correct formula for the harmonic mean

In option (b), we are given the correct formula for the harmonic mean of two numbers a and b: $$\mathrm{HM}=\frac{2 \times \mathrm{ab}}{\mathrm{a}+\mathrm{b}}$$

Identify the values of 'a' and 'b'

In option (c), we find the values of a and b. Here, a = 5 and b = 3.

Apply the formula with the given values

Using option (a), we substitute the values of a and b in the formula: $$\mathrm{HM}=\frac{2 \times 5 \times 3}{5+3}$$

Calculate the harmonic mean

Use option (d) to calculate the harmonic mean: $$\mathrm{HM}=\frac{30}{8}=\frac{15}{4}$$ The correct sequence of steps for solving this exercise is option (2), which is bcad.

Key concepts

Arithmetic Progression

An arithmetic progression (AP) is one of the most fundamental concepts in the study of sequences and series. It is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference, denoted as 'd'. For example, in the sequence 2, 5, 8, 11, ..., the common difference is 3 because each term after the first is 3 units greater than the preceding term.

Formally, an arithmetic progression with the first term 'a' and the common difference 'd' can be expressed as: \[ a, a + d, a + 2d, a + 3d, ... \].

Understanding arithmetic progressions is crucial because they appear frequently in various mathematical contexts and real-world situations, such as predicting the nth term of a given sequence, summing the terms of a sequence, and even in financial calculations involving simple interest.

Sequences and Series

Sequences and series are the bedrock of many mathematical concepts and appear extensively in fields such as physics, economics, and engineering. A sequence is a set of numbers listed in a specific order. The numbers in the sequence are called terms. If you sum the terms of a sequence, you get a series.

For example, if you have a sequence of numbers 1, 2, 3, 4, ..., the corresponding series would be the sum of these numbers, which could continue indefinitely or stop at a particular term.

There are different types of series, such as the finite series where the number of terms is limited, and the infinite series where the series continues without end. Arithmetic and geometric series are probably the most well-known types, each with unique characteristics and formulas. The former involves a sum of numbers with a common difference, while the latter involves a sum of numbers with a common ratio.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (addition, subtraction, multiplication, and division). For instance, \( 3x^2 - 2xy + 5y - 7 \) is an algebraic expression. In the context of sequences and series, algebraic expressions can define the nth term, or they can be used to represent the sum of the terms in a series.

An important aspect of working with algebraic expressions is understanding how to simplify them and solve equations or inequalities they may form. This involves skills such as combining like terms, using distributive properties, and applying various algebraic identities. Algebraic expressions play a pivotal role in the solving of a wide range of problems across mathematics.