Question

If \(\mathrm{a}_{\mathrm{r}}>0, \mathrm{r} \in \mathrm{N}\) and \(\mathrm{a}_{1} \mathrm{a}_{2} \ldots \ldots \mathrm{a}_{2 \mathrm{~m}}\) are in \(\mathrm{AP}\), then, (a) \(\mathrm{n}-1\) (b) \(\frac{n\left(a_{1}+a_{2 n}\right)}{\sqrt{a_{1}}+\sqrt{a_{n+1}}}\) (c) \(\frac{n-1}{\sqrt{a n}+\sqrt{a n+1}}\) (d) None of these

Short answer

In this exercise, we found the expression for the sum of the first n terms of an arithmetic progression with a common difference d and positive terms as follows: $$ S_n = \frac{n(a_1 + a_{n+1})}{2} $$ When comparing this expression to the given options, none of them match our result. Therefore, the correct answer is option (d) None of these.

Step-by-step solution

Find the general term of the AP

We are given that \(a_r > 0\) for all \(r \in N\) and \(a_1, a_2, ..., a_{2n}\) are in AP. Let's define a common difference for the AP as \(d\). So the general term of the AP, \(a_r\), can be expressed as: $$ a_r = a_1 + (r - 1)d $$ where \(r \in N\).

Write the terms till \(a_{2n}\)

Let's list down the terms till \(a_{2n}\) using the general term expression we have derived. $$ a_1 = a_1 $$ $$ a_2 = a_1 + d $$ $$ a_3 = a_1 + 2d $$ Continuing in the sequence, we can write the term \(a_{2n}\) as: $$ a_{2n} = a_1 + (2n - 1)d $$

Calculate the sum of first n terms, \(S_n\)

We will now calculate the sum of first n terms, \(S_n\), by adding the terms \(a_1, a_3, a_5,..., a_{2n-1}\) The sum of first n terms will be: $$ S_n = a_1 + a_3 + a_5 + ... + a_{2n-1} $$ We can rewrite the terms using the expressions derived in Step 2: $$ S_n = a_1 + (a_1 + 2d) + (a_1 + 4d) + ... + (a_1 + (2n-1)d) $$

Simplify \(S_n\) expression

Now, let's simplify the sum \(S_n\) as follows: $$ S_n = na_1 + 2d(1+2+3+...+(n-1)+n) $$ Using the formula for the sum of first n natural numbers, \(\frac{n(n+1)}{2}\), we can write: $$ S_n = na_1 + 2d\frac{n(n+1)}{2} $$ Now, let's simplify further: $$ S_n = na_1 + dn(n+1) = \frac{n[(n+1)d + 2a_1]}{2} $$ We want to express \(S_n\) in the given options format. Therefore, let's try to rewrite \(S_n\) expression in terms of \(a_n\) and \(a_{n+1}\). Notice that \(a_1 + 2a_1 = a_{n+1}\) and \(a_{n+1} = a_1 + (n+1)d\). Thus, we can rewrite \(S_n\) as: $$ S_n = \frac{n(a_1 + a_{n+1})}{2} $$ Now, let's compare the given options with our result: (a) \(n-1\) -> this is not equal to our result. (b) \(\frac{n(a_1 + a_{2n})}{\sqrt{a_1}+ \sqrt{a_{n+1}}}\) -> this is not equal to our result. (c) \(\frac{n-1}{\sqrt{a n}+\sqrt{a n+1}}\) -> this is not equal to our result. (d) None of these -> Since none of the given options match with our result, the correct answer is option (d) None of these.

Key concepts

Common Difference

In an arithmetic progression (AP), the difference between any two consecutive terms is known as the Common Difference. This is a crucial aspect of an AP and is denoted by \(d\). It helps in generating the sequence, by adding it to the previous term to get the next one.
For example, if the first term \(a_1\) is 3 and the common difference \(d\) is 2, the sequence begins as 3, 5, 7, 9, and so forth.
The common difference can be calculated by subtracting any term from its succeeding term, i.e., \(d = a_{r+1} - a_r\). Understanding \(d\) allows us to easily predict or calculate any term in the AP.

General Term

The General Term of an arithmetic progression is an expression that allows us to find any term in that sequence without listing all preceding terms. This is represented by \(a_r\), where \(r\) is the position of the term in the sequence.
The formula for the general term is often written as:

• \(a_r = a_1 + (r - 1)d\)

Here, \(a_1\) is the first term of the sequence, \(r\) is the term number, and \(d\) is the common difference.
This formula is essential because it gives you direct access to any term in the sequence, making it easy to compute specific terms without needing to go term by term.

Natural Numbers

Natural numbers are the set of positive integers starting from 1, 2, 3, and continuing onwards indefinitely. They are sometimes denoted by \(\mathbb{N}\). These numbers are the building blocks of algebra and arithmetic sequences.
When working with arithmetic progressions, natural numbers often serve as positions (\(r\)) when calculating terms using the general term formula, as they represent the count or index of terms in a sequence.
In the context of the problem involving arithmetic progression, each term \(a_r\) corresponds to a distinct natural number \(r\), indicating the position of that term within the sequence.

Sum of Series

The Sum of a series in an arithmetic progression refers to the total of a consecutive block of terms from the progression. Calculating this is simple with the right formula, and it is a frequent problem-solving technique in mathematics.
The sum of the first \(n\) terms, denoted \(S_n\), can be computed using the formula:

• \(S_n = \frac{n}{2} \times (a_1 + a_n)\)
• Alternatively, \(S_n = n \times \frac{[2a_1 + (n-1)d]}{2}\)

This formula shows that the sum is the product of the number of terms and the average of the first and last term of the segment being summed.
In the problem, simplifying the expression of \(S_n\) was critical to determining whether any answer options matched the given conditions.