For an A. P., \(S_{100}=3 S_{50}\). The value of \(S_{150}: S_{50}=\) (A) 8 (B) 3 (C) 6 (D) 10

Understanding the sum of an arithmetic series is a fundamental concept in mathematics. In an arithmetic series, we deal with a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. The sum of the first 'n' terms of an arithmetic progression (AP) can be found using the formula:
\[\begin{equation}S_n = \frac{n}{2}(a_1 + a_n)\end{equation}\]
where:

• \(S_n\) is the sum of the first 'n' terms,
• \(a_1\) is the first term, and
• \(a_n\) is the nth term of the sequence.

An alternative version of the formula uses the first term and the common difference 'd':
\[\begin{equation}S_n = \frac{n}{2}(2a_1 + (n-1)d)\end{equation}\]
This formula is derived by multiplying the average of the first and last term by the number of terms. In a more practical sense, this enables students to easily calculate the sum of terms in a sequence without having to add each individually—especially useful as the number of terms grows large.

When it comes to arithmetic sequences, there's a direct way to find any term in the sequence without listing all the preceding terms. This is incredibly efficient as it saves time and provides a quick method for finding large terms in a sequence. The nth term of an arithmetic sequence is given by the formula:
\[\begin{equation}a_n = a_1 + (n-1)d\end{equation}\]
where:

• \(a_n\) is the nth term we want to find,
• \(a_1\) is the first term of the sequence, and
• \(d\) is the common difference between the terms.

With this formula, if you know the first term and the common difference of the arithmetic sequence, you can determine any term in the sequence. This is a cornerstone in understanding arithmetic progression and is invaluable for solving problems involving sequences.

Comparing the sums of arithmetic series can sometimes involve finding the ratio of these sums. Ratios give us a relative measure of two quantities and, in the context of arithmetic series, allow us to compare the sums of different numbers of terms without needing to calculate the actual sums. In the exercise, we're asked to find the ratio of the sum of the first 150 terms of an arithmetic series to the sum of the first 50 terms. The relationship between the terms, represented by a ratio, can reveal patterns and provide insights into the progression as a whole. For example, if the sums are in a simple ratio, this can indicate a linear relationship between the number of terms and the sum.

Practice with Exercise

The given exercise problem leverages this understanding by providing a specific ratio between different portions of the series and then asking for the ratio of sums over different intervals. The solution involves using the sum formulas of an arithmetic series and simplifies into a ratio that offers a more straightforward comparison between the sums. This consolidates the concept that relationships within arithmetic progressions can be analyzed and understood through the lens of ratios, offering a more accessible entry point into series sums for students.