Question

If \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) are in arithmetic progression, then \(\mathrm{b}+\mathrm{c}, \mathrm{c}+\mathrm{a}\) and \(\mathrm{a}+\mathrm{b}\) are in (1) Arithmetic Progression (2) Geometric Progression (3) Harmonic Progression (4) None of these

Short answer

Answer: (1) Arithmetic Progression

Step-by-step solution

Identify the arithmetic progression

Given that a, b, and c are in arithmetic progression, we know there exists a common difference (d) such that b = a + d and c = a + 2d.

Calculate the sums b + c, c + a, and a + b

Using the substitutions from Step 1, we will calculate the sums: 1. b + c = (a + d) + (a + 2d) = 2a + 3d 2. c + a = (a + 2d) + a = 2a + 2d 3. a + b = a + (a + d) = 2a + d

Determine the relationship between the sums

The three sums we found in Step 2 are: 1. 2a + 3d 2. 2a + 2d 3. 2a + d Now, we can check if they form an arithmetic, geometric, or harmonic progression, or none of these.

Check for Arithmetic Progression

To be an arithmetic progression, the common difference should be the same between each term. We can see that the difference between each term is d, which is constant. Therefore, the sums are in Arithmetic Progression. The answer is (1) Arithmetic Progression.

Key concepts

Common Difference

In an arithmetic progression (AP), the 'common difference' is one of the most critical aspects that characterizes the sequence. It is the constant value that, when added to any term of the sequence, gives us the next term. To further understand this concept, consider the arithmetic progression consisting of numbers 3, 5, 7, 9, and so on. Here, the common difference is 2, because each term is 2 units greater than the term before it.

To mathematically express the common difference, we often use the symbol 'd'. If we have an arithmetic progression with terms 'a', 'a+d', 'a+2d', and so on, the common difference is the value 'd'. This 'd' can be positive, negative, or even zero. In the case that 'd' is zero, every term in the sequence is the same. This simple concept ensures a uniform progression throughout the sequence and allows for straightforward calculations when dealing with arithmetic progressions.

Arithmetic Sequence

An 'arithmetic sequence' is a specific type of number sequence wherein each term after the first is obtained by adding the common difference to the preceding element. The formula to find the nth term of an arithmetic sequence is given by
\[\begin{equation}a_n = a_1 + (n - 1)d\text{, where}\begin{align*} a_n &\text{ is the nth term}, a_1 &\text{ is the first term}, n &\text{ is the term number}, d &\text{ is the common difference}. \text{\end{align*}}\text{\end{equation}\]}For instance, in an arithmetic sequence such as 2, 4, 6, 8, ..., to find the 5th term, we use the formula mentioned above: \[\begin{equation}a_5 = 2 + (5 - 1) \times 2 = 2 + 8 = 10.\text{\end{equation}\]}

Properties of an Arithmetic Sequence

• The sequence progresses uniformly by the common difference.
• The graph of an arithmetic sequence will form a straight line.
• The arithmetic mean between any two terms in the sequence will always equal the term that is exactly in the middle of those two.

Understanding the arithmetic sequence is crucial for recognizing patterns and solving problems that involve regular intervals, such as the problem given in this exercise.

Mathematical Series

A 'mathematical series' refers to the sum of the elements of a sequence. When the sequence is arithmetic, the resultant series is called an arithmetic series. The sum of the first 'n' terms of an arithmetic sequence can be found using the formula:\[\begin{equation}S_n = \frac{n}{2}(2a_1 + (n - 1)d)\text{, where}\begin{align*} S_n &\text{ is the sum of the first } n \text{ terms}, a_1 &\text{ is the first term}, d &\text{ is the common difference}, n &\text{ is the number of terms to be added.}\text{\end{align*}}\text{\end{equation}\]}For example, if you have the arithmetic sequence 3, 5, 7, 9 and wish to find the sum of the first 4 terms, the formula yields:\[\begin{equation}S_4 = \frac{4}{2}(2 \times 3 + (4 - 1) \times 2) = 2(6+6) = 24.\text{\end{equation}\]}Understanding how to work with mathematical series is fundamental not only in pure mathematics but also in practical applications such as finance, computer science, and engineering, where cumulative quantities and successive intervals play an important role.