Question

If \(n\) arithmetic means are inserted between two quantities \(a\) and \(b\), then their sum is equal to : (a) \(n(a+b)\) (b) \(\frac{n}{2}(a+b)\) (c) \(2 n(a+b)\) (d) \(\frac{\pi}{2}(a-b)\)

Short answer

Answer: The sum of \(n\) arithmetic means inserted between two quantities \(a\) and \(b\) is \(\frac{n}{2}(a+b)\).

Step-by-step solution

Understand arithmetic mean

An arithmetic mean is a value that lies between two numbers and is the average of those two numbers. When we insert \(n\) arithmetic means between two quantities \(a\) and \(b\), it means we create a sequence of \(n+2\) terms, where the first term is \(a\), the last term is \(b\), and the first arithmetic mean is the second term, and so on.

Formulate the equation for the sum of \(n\) arithmetic means

Let's denote the sum of the \(n\) arithmetic means as \(S\). We know that the sum of all terms in an arithmetic sequence is given by the formula \(S = \frac{n}{2}(a + b)\), where \(n\) is the number of terms, \(a\) is the first term, and \(b\) is the last term. But in our case, we have a sequence of \(n+2\) terms (with \(a\) and \(b\) included). So, the sum of all terms (\(n\) arithmetic means, \(a\) and \(b\)) is: \(S_{total} = \frac{n+2}{2}(a+b)\)

Find the sum of \(n\) arithmetic means without \(a\) and \(b\)

Now, we need to find the sum of only the \(n\) arithmetic means without \(a\) and \(b\). To do this, we subtract the sum of \(a\) and \(b\) from the total sum (\(S_{total}\)): \(S = S_{total} - (a+b)\) Substituting \(S_{total}\) from Step 2: \(S = \frac{n+2}{2}(a+b) - (a+b)\)

Simplify the equation and compare it with the given options

Now, let's simplify the equation: \(S = \frac{n+2}{2}(a+b) - \frac{2(a+b)}{2}\) \(S = \frac{n(a+b)}{2}\) After simplification, we get that the sum of \(n\) arithmetic means is \(\frac{n(a+b)}{2}\). Comparing this with the given options, we can see that it matches option (b). Hence, the correct answer is: (b) \(\frac{n}{2}(a+b)\)

Key concepts

Sum of Arithmetic Means

When we talk about the sum of arithmetic means, we are discussing the total of several numbers that are equally spaced between two values, known as quantities. If you think of arithmetic means as stepping stones between two numbers, you're on the right track. In a math problem, if you have two endpoints, say \( a \) and \( b \), and you want to find the sum of all the means inserted between these points, there's a neat formula to simplify your task.

• The sum involves adding up all these means together.
• This scenario requires the means to be equally spaced, forming what's called an arithmetic sequence.

To calculate, first determine the total sequence including \( a \) and \( b \), which equals \( n + 2 \), where \( n \) is the number of inserted means. Using the formula for the sum of an arithmetic sequence, we start with\[S_{total} = \frac{n+2}{2}(a+b) \]However, our task is to consider only the inserted arithmetic means, not the entire sequence. By removing the sum of \( a \) and \( b \) from \( S_{total} \), we arrive at the sum of the inserted means:\[S = \frac{n(a+b)}{2} \]

Arithmetic Sequence

An arithmetic sequence is a formation where each number or term after the first is obtained by adding a fixed amount – usually called the common difference \( d \) – to the previous term. To build an arithmetic sequence, you start with a number, commonly termed as the first term \( a \), and continuously add this common difference. This sequence forms a regular pattern that is easy to predict.

• A sequence like \( a, a+d, a+2d, \ldots \) is a simple example.
• The common difference \( d \) is uniform throughout the sequence.
• The sequence continues in both directions without end.

When arithmetic means are inserted between two numbers, those means form an arithmetic sequence. Each term steadily progresses by the same amount until reaching \( b \). This is particularly important when solving problems involving such sequences, as it helps in calculating their sum or other properties effectively.

Inserted Means Between Quantities

Inserted means or commonly called arithmetic means, are numbers that fill the gap between two distinct values \( a \) and \( b \) such that they form a smooth sequence. Specifically in math, when we "insert means," we're adding values that evenly cover the space between two endpoints.

• Let's say you have two numbers, \( a \) and \( b \).
• If you insert \( n \) arithmetic means between these two numbers, you're creating a new set of numbers: \( a, M_1, M_2, \ldots, M_n, b \).
• This lengthens into an arithmetic sequence with \( n+2 \) terms.

The key to understanding this is recognizing how these means act as bridges that equalize the difference between \( a \) and \( b \). This allows for practical applications, such as in financial modeling or statistical analysis, where understanding the spread or distribution between data points is crucial. To get the arithmetic means, knowing either the total sum or the common difference could be your best approach, as each mean increases uniformly.