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Chapter 14: Problem 41

Find the sum of the even numbers from 2 to 100 , inclusive.

Short Answer

Expert verified
The sum of the even numbers from 2 to 100 is 2550.

Step by step solution

01

- Identify the Sequence

The sequence of even numbers from 2 to 100 is an arithmetic sequence. The first term (a) is 2, the common difference (d) is 2, and the last term (l) is 100.
02

- Determine the Number of Terms

Use the formula for the nth term of an arithmetic sequence: \( a_n = a + (n-1) \times d \). Set \( a_n \) equal to 100 and solve for n: \( 100 = 2 + (n-1) \times 2 \).Rearrange to find n: \( 100 = 2n \) so \( n = 50 \).
03

- Use the Sum Formula for an Arithmetic Sequence

The sum of the first n terms of an arithmetic sequence is given by the formula: \[ S_n = \frac{n}{2} \times (a + l) \].Substitute the values: \( S_{50} = \frac{50}{2} \times (2 + 100) \).Calculate to find the sum: \( S_{50} = 25 \times 102 = 2550 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

arithmetic sequence
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference, often denoted by d. For example, in the sequence 2, 4, 6, 8, ..., the common difference is 2. Each term is found by adding 2 to the previous term.
To identify an arithmetic sequence, check if the differences between consecutive terms are the same. If they are, it's an arithmetic sequence.
Arithmetic sequences have a straightforward structure, which makes them easy to work with, especially when computing sums or finding specific terms in the sequence.
sum formula
The sum of an arithmetic sequence can be calculated using a specific sum formula. This formula is highly useful when you want to find the sum of a large number of terms without adding them individually.
The sum formula for the first n terms of an arithmetic sequence is:
\( S_n = \frac{n}{2} \times (a + l) \)
Here, \( S_n \) is the sum of the first n terms, n is the number of terms, a is the first term, and l is the last term.
Using this formula simplifies calculations and saves time. For example, to find the sum of the even numbers from 2 to 100, you first identify the sequence, determine the number of terms, and then plug those values into the sum formula.
even numbers
Even numbers are integers that are divisible by 2. They always end in 0, 2, 4, 6, or 8. Examples of even numbers include 2, 4, 6, 8, ...
When dealing with sequences of even numbers, they often form an arithmetic sequence with a common difference of 2. For instance, the sequence of even numbers from 2 to 100 has a first term (a) of 2 and a common difference (d) of 2.
To find the sum of even numbers in a range, you can follow the method detailed earlier: identify the sequence, determine the number of terms, and apply the sum formula. This method is efficient and avoids manual addition of individual terms.

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