Calculate Sum: ∑(r+2) From R=1 To 50

Hey guys! Ever stumbled upon a mathematical expression that looks intimidating but is actually quite fascinating? Today, we're going to unlock the secrets of one such expression: the summation $\sum_{r=1}^{50}(r+2)$. This might look like a bunch of symbols at first glance, but trust me, it's a journey into the beautiful world of arithmetic series and summations. So, grab your thinking caps, and let's dive in!

Decoding the Summation Notation

Before we jump into solving, let's break down what this notation actually means. The symbol $\sum$ (Sigma) is the Greek capital letter S, which in mathematics stands for sum. The expression below the Sigma, r=1, tells us where our summation starts. In this case, we begin with r equal to 1. The number above the Sigma, 50, indicates where our summation ends. So, we'll be plugging in values for r from 1 all the way up to 50. The expression (r+2) is what we're actually summing up for each value of r. Think of it as a formula that generates a series of numbers, and we're adding all those numbers together.

To put it simply, $\sum_{r=1}^{50}(r+2)$ means:

(1 + 2) + (2 + 2) + (3 + 2) + ... + (50 + 2)

We are adding the numbers starting from when r is 1 to when r is 50. Each time we increment r we add 2 to it, and add it to the total. It might seem like a tedious task to calculate this manually, but fear not! We have some clever mathematical tools at our disposal. We can break this summation down into simpler components. The key here is to recognize that we're dealing with an arithmetic series, or rather, a sum of two arithmetic series. One is a constant and one is a simple arithmetic progression. This is a crucial step in simplifying the problem and making it manageable.

The Power of Arithmetic Series

Now that we understand the notation, let's talk strategy. The expression (r+2) can be viewed as the sum of two separate terms: r and 2. This allows us to split the original summation into two simpler summations:

$\sum_{r=1}^{50}(r+2) = \sum_{r=1}^{50}r + \sum_{r=1}^{50}2$

This seemingly small step is actually a major breakthrough. It allows us to leverage known formulas for arithmetic series and constant summations. The first summation, $\sum_{r=1}^{50}r$, represents the sum of the first 50 natural numbers. This is a classic arithmetic series, and there's a nifty formula to calculate it directly. The second summation, $\sum_{r=1}^{50}2$, is even simpler. We're just adding the number 2 to itself 50 times. Think of it like repeatedly adding the same number which is akin to multiplication.

The formula for the sum of the first n natural numbers is:

$\sum_{r=1}^{n}r = \frac{n(n+1)}{2}$

This formula is a cornerstone of arithmetic series and a powerful tool to have in your mathematical arsenal. It saves us the trouble of manually adding each number, especially when dealing with large values of n. The beauty of this formula lies in its elegance and efficiency. It encapsulates a pattern that has been observed and proven over centuries.

For our case, n = 50, so we can plug that value into the formula:

$\sum_{r=1}^{50}r = \frac{50(50+1)}{2} = \frac{50 * 51}{2} = 25 * 51 = 1275$

So, the sum of the first 50 natural numbers is 1275. We've conquered the first part of our problem! Now, let's tackle the second summation.

Constant Summations: A Breeze

The second summation, $\sum_{r=1}^{50}2$, is a constant summation. We're simply adding the constant value 2 a total of 50 times. This is equivalent to multiplying 2 by 50:

$\sum_{r=1}^{50}2 = 2 * 50 = 100$

See? That was super easy! Constant summations are generally straightforward and require minimal calculation. They represent a fundamental concept in mathematics: repeated addition. In essence, we are performing multiplication by repeatedly adding the same value. This simple operation forms the basis for more complex mathematical concepts.

Putting it All Together

Now that we've calculated both individual summations, we can combine them to find the final answer. Remember, we split the original summation as follows:

$\sum_{r=1}^{50}(r+2) = \sum_{r=1}^{50}r + \sum_{r=1}^{50}2$

We found that:

$\sum_{r=1}^{50}r = 1275$

$\sum_{r=1}^{50}2 = 100$

Therefore:

$\sum_{r=1}^{50}(r+2) = 1275 + 100 = 1375$

And there you have it! The sum of the series is 1375. We've successfully navigated the summation notation, applied the arithmetic series formula, and handled a constant summation. Great job, everyone!

Why This Matters: Applications and Beyond

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