# Solving the Mystery: What is the Following Quotient?

As we delve deeper into the realms of mathematics, we often come across a variety of intriguing problems that keep us hooked. Some are easy to solve while others may require layers of deductions and calculations. However, one such problem that has puzzled many mathematicians over time is deciphering the following quotient. In this article, we will attempt to solve this mystery and provide you with a detailed insight into what it entails.

Before we dive deep into the complexities of this problem, let us first understand what a quotient is. A quotient is simply the result of dividing one number by another. For instance, if we have 10 apples and want to divide them equally amongst two people, the quotient would be 5 (10 divided by 2). Similarly, if we divide 8 by 4, the quotient would be 2.

Now, coming back to the main topic of discussion, the following quotient is a problem that has been around for a long time. The quotient is usually represented as follows:

(1 + 2 + 3 + … + n) / n

Here, the dots indicate the continuation of the sequence until the nth term. For instance, if n = 6, the sequence would look like:

1 + 2 + 3 + 4 + 5 + 6

To find the solution to this problem, we need to evaluate the sum of the sequence on the numerator and divide it by n.

Now, let us take a step back and explore the history of this quotient. The origin of this problem dates back to the 18th century when the famous mathematician, Leonhard Euler, came up with a formula for this problem. He observed that the sum of the first n numbers can be calculated using the formula (n * (n + 1)) / 2. Using this formula, he derived the solution for the following quotient as follows:

((n * (n + 1)) / 2) / n

Simplifying this equation, we get:

((n * (n + 1)) / 2n)

Here, we can notice that n and (n + 1) have a common factor of 2. We can simplify further by dividing both the numerator and denominator by 2. This gives us:

((n + 1) / 2)

Hence, the solution to the following quotient is ((n + 1) / 2).

Now that we have explored the history of this problem and obtained a solution, let us take a closer look at how we can prove Euler’s formula. To do so, we will use mathematical induction.

Firstly, we will assume the formula for n = 1. Here, the sequence would be:

1

Substituting this value in Euler’s formula, we get:

((1 * (1 + 1)) / 2)

Simplifying this equation, we get:

(2 / 2)

Which is equal to 1. Hence, the formula holds true for n = 1.

Next, we will assume that the formula holds true for an arbitrary value k. This means that:

(1 + 2 + 3 + … + k) = (k * (k + 1)) / 2

Now, we will prove that the formula also holds true for (k + 1). This means that:

(1 + 2 + 3 + … + k + (k + 1)) = ((k + 1) * ((k + 1) + 1)) / 2

To do so, we can simply add (k + 1) to both sides of the equation for k. This gives us:

(1 + 2 + 3 + … + k + (k + 1)) = (k * (k + 1)) / 2 + (k + 1)

Further simplifying this equation, we get:

(1 + 2 + 3 + … + k + (k + 1)) = ((k + 1) * ((k + 1) + 1)) / 2

Hence, we have successfully proven Euler’s formula using mathematical induction.

In conclusion, solving the mystery of the following quotient required us to understand what a quotient is and delve into the history of this problem. We saw how Leonhard Euler used his formula to derive a solution and explored the steps required to prove it using mathematical induction. This problem may seem simple at first glance, but it holds immense importance in the field of mathematics and has played a significant role in shaping it over time.