Problem 55 - Precalculus with Limits

Question

In Exercises 51 - 58, find the sum of the finite arithmetic sequence. Sum of the first \( 50 \) positive even integers

Step-by-Step Solution

Answer

The sum of the first 50 even positive integers is 2550.

Step 1: Identify the elements of the sequence

This arithmetic sequence starts with a first term (a) of 2. It is an arithmetic sequence of positive even numbers, so the common difference (d) is 2. The number of terms (n) to be added is 50.

Step 2: Find the last term of the sequence

The nth term of an arithmetic sequence can be found by: \[ l = a + (n - 1) * d \] So the last term (l) is \( 2 + (50 - 1)*2 = 100 \)

Step 3: Apply the arithmetic sum formula

Use the formula for sum of an arithmetic sequence, \[ S = \frac{n}{2} (a + l) \] Substituting the known values, \( S = \frac{50}{2} * (2 + 100) = 2550\)

Key Concepts

Sum of Arithmetic Sequences

When you want to find the sum of an arithmetic sequence, you need to know a simple formula. This formula will help you add up all the terms in the sequence quickly and accurately. An arithmetic sequence is simply a list of numbers where each number increases by a constant amount — that's our **common difference**.

For example, in our exercise, we're adding the first 50 positive even integers like 2, 4, 6, and so on. To calculate the sum of such a sequence, use the sum formula:

• The formula is: \[ S = \frac{n}{2} (a + l) \]
• **S** is the sum of the sequence.
• **n** is the number of terms.
• **a** is the first term.
• And **l** is the last term.

By filling these values into the formula, you neatly find out how much all those numbers add up to. In our example, this comes out as 2550 — it’s that easy with the right steps!

Common Difference

The concept of a "common difference" is fundamental when dealing with arithmetic sequences. The common difference is what makes an arithmetic sequence different from a random group of numbers.

It is the constant amount added to each term to get the next term. In simple terms, it's how much you're adding each time.

• In our sequence of positive even integers, the first term is 2.
• The next term is 4 because we've added 2.
• We continue this pattern: 2, 4, 6, 8, and so on with a constant difference of 2.

Knowing the common difference is crucial as it allows us to find any term in the sequence quickly and determines how the sequence "grows." As seen in the solution, the common difference helps calculate the last term when it was needed for the sum.

Arithmetic Sequence Formula

The arithmetic sequence formula is one of the basic tools you'll use when dealing with such sequences. It provides a way to find any term in the sequence, especially important when you don't have specific terms but need to calculate or verify based on position.

The formula is given by:

• \[ l = a + (n - 1) \cdot d \]
• Where **l** is the term you’re looking for, which might be the last term if you reach the end of your sequence.
• **a** is the first term of your sequence.
• **d** is the common difference, which we talked about earlier.
• **n** is the number of the term you want to find.

In our example, we used this formula to find the 50th (or last) term, which came out to be 100. Knowing each component of the formula means you can tackle any arithmetic sequence with confidence!