Problem 67
Question
In Exercises 67 - 74, find the partial sum. \( \sum_{n=1}^{50}n \)
Step-by-Step Solution
Verified Answer
The partial sum of the first 50 natural numbers is 1275.
1Step 1: Understand the problem
The given expression \( \sum_{n=1}^{50}n \) represents the sum of the first 50 natural numbers. We need to calculate this sum.
2Step 2: Identify the relevant formula
The formula to calculate the sum of the first N natural numbers is \( \frac{N*(N+1)}{2} \).
3Step 3: Apply the formula
We substitute N=50 into the formula, giving us \( \frac{50*(50+1)}{2} \)
4Step 4: Calculate the result
Solving the above expression, we get \( \frac{50*51}{2} = 1275 \)
Key Concepts
Partial SumNatural NumbersSum Formula
Partial Sum
A partial sum is essentially the sum of the first few terms of a sequence or series. It does not include all terms possible, hence the name "partial." In mathematical problems and exercises, the concept of partial sum is crucial to understand how to evaluate sums where the full series is not required or when computing just a segment of it.
When working with series, you'll find partial sums written using the summation symbol (Σ), denoting that you only sum the terms from the first up to a specific number. In our example, the notation \( \sum_{n=1}^{50} n \) represents a partial sum from 1 to 50.
Partial sums are especially useful in various branches of mathematics, such as calculus and number theory. For example, they help estimate the total of infinite series, by providing a finite approximation by stopping at a certain point.
When working with series, you'll find partial sums written using the summation symbol (Σ), denoting that you only sum the terms from the first up to a specific number. In our example, the notation \( \sum_{n=1}^{50} n \) represents a partial sum from 1 to 50.
Partial sums are especially useful in various branches of mathematics, such as calculus and number theory. For example, they help estimate the total of infinite series, by providing a finite approximation by stopping at a certain point.
Natural Numbers
Understanding natural numbers is crucial before jumping into summation problems. Natural numbers are the set of positive integers beginning from 1, and they progress infinitely as 1, 2, 3, 4, and so on. These numbers are basic counting numbers that we often learn first in mathematics.
Most often in summation problems, especially those involving partial sums, natural numbers play a significant role. It's these numbers that are typically summed in series like \( \sum_{n=1}^{50} n \), which helps lead us to formulas and procedures of calculating sums.
- They are represented mathematically by \( \mathbb{N} \).
- Natural numbers do not include negative numbers or zero.
- They are used to describe quantities in real-world counting scenarios.
Most often in summation problems, especially those involving partial sums, natural numbers play a significant role. It's these numbers that are typically summed in series like \( \sum_{n=1}^{50} n \), which helps lead us to formulas and procedures of calculating sums.
Sum Formula
The sum formula for natural numbers is a powerful tool for quickly finding the sum of a sequence of consecutive natural numbers without adding each number manually. This formula can be expressed as:\[S = \frac{N \times (N+1)}{2}\]Where:
Applying this formula allows you to calculate the partial sum efficiently. For example, if you wish to determine the sum of the first 50 natural numbers, substitute \( N = 50 \) into the formula: \[ S = \frac{50 \times 51}{2} \]
This results in 1275, providing a quick and straightforward way to calculate without manual addition.
The sum formula highlights the structure and pattern within natural numbers, and reveals elegant relationships in mathematics, simplifying complex arithmetic into simple multiplication and division.
- \( S \) is the sum of natural numbers.
- \( N \) is the last natural number in the sequence you're summing up to.
Applying this formula allows you to calculate the partial sum efficiently. For example, if you wish to determine the sum of the first 50 natural numbers, substitute \( N = 50 \) into the formula: \[ S = \frac{50 \times 51}{2} \]
This results in 1275, providing a quick and straightforward way to calculate without manual addition.
The sum formula highlights the structure and pattern within natural numbers, and reveals elegant relationships in mathematics, simplifying complex arithmetic into simple multiplication and division.
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