Finding the Common Difference in a Series Given the Sum of the First 300 Terms

In this article, we will explore the concept of an arithmetic progression (AP) where the first term is 12 and the sum of the first 300 terms equals 300. We will delve into how to find the common difference in such a series, providing various solutions and even considering non-integer solutions.

Understanding the Problem

The problem at hand states that the first term of an arithmetic progression (AP) is 12, and the sum of the first 300 terms of the series is 300. We are tasked with finding the common difference of this series. Let's break down the solution step by step.

Arithmetic Progression Basics

An arithmetic progression (AP) is a sequence of numbers such that the difference between any two successive members is constant. The general form of an AP is given by:

an a (n-1)d

where:

an - the nth term

a - the first term

d - the common difference

n - the term number

The Sum of the First 300 Terms

The sum of the first n terms of an AP is given by the formula:

S_n n/2 [2a (n-1)d]

In this question, we have:

S_300 300

Plugging in the values, we get:

300 300/2 [2(12) (300-1)d]

Simplifying this:

300 150 [24 299d]

Dividing both sides by 150:

2 24 299d

299d -22

d -22/299

Now, let's consider some alternative approaches and solutions.

Variations and Solutions

Given the flexible nature of the problem, there are multiple solutions to the problem. Let's look at some of these:

Series: 12, 288... with common difference 276. This series can be represented as:
12, 12 276, 12 2*276, ..., which does not meet the sum condition.

Series: 12, 100, 188... with common difference 88. This series can be represented as:
12, 12 88, 12 2*88, ..., which does not meet the sum condition.

Series: 12, 54, 96, 138... with common difference 42. This series can be represented as:
12, 12 42, 12 2*42, ..., which does not meet the sum condition.

Series: 12, 36, 60, 84, 108... with common difference 24. This series can be represented as:
12, 12 24, 12 2*24, ..., which does not meet the sum condition.

Series: 12, 27.2, 42.4, 57.6, 72.8, 88... with common difference 15.2. This series can be represented as:
12, 12 15.2, 12 2*15.2, ..., which does not meet the sum condition.

Series: 12, 22 2/7, 32 4/7, 42 6/7, 53 1/7, 63 3/7, 73 5/7... with common difference 10 2/7. This series can be represented as:
12, 12 10 2/7, 12 2*10 2/7, ..., which does not meet the sum condition.

Each of these series represents different common differences, and none of them meet the exact sum condition when n 300. However, this scenario provides a good exercise in understanding arithmetic progressions.

Final Solution

The most straightforward solution, considering the sum of the first term is 300, is:

12, 16, 20, 24, 28, 32, 36, 40, 44, 48, ...

This series has a common difference of 4, and we can verify this:

S_300 300/2 [2*12 (300-1)*4] 300/2 [24 1196] 300 [610] 300

This meets the exact sum of 300 given.

Therefore, the common difference d 4.

Conclusion

The problem of finding the common difference in an arithmetic progression where the sum of the first 300 terms is 300 can be approached in multiple ways. While the provided series do not meet the exact sum condition, the final solution showcases a straightforward and accurate method to find the common difference.

Key Takeaways:

The sum of the first n terms of an AP can be calculated using the formula S_n n/2 [2a (n-1)d].

Flexibility in arithmetic progressions allows for multiple common differences that could satisfy the initial conditions.

The solution provided here is based on the most straightforward and accurate series.