Problem 48 - College Algebra

Question

Write out the first three terms and the last term. Then use the formula for the sum of the first \(n\) terms of an arithmetic sequence to find the indicated sum. $$\sum_{i=1}^{4}(-2 i+6)$$

Step-by-Step Solution

Answer

So the first three terms of the sequence are \(4, 2, 0\) and the fourth (last) term is \(-2\). The sum of the first four terms of the sequence is \(4\).

Step 1: Identify the first three terms

The first three terms of the arithmetic sequence can be found by substituting \(i=1\), \(i=2\) and \(i=3\) into the equation \(-2i+6\). This will give the terms as follows:\nTerm 1 when \(i=1\) is \(-2(1)+6 = 4)\nTerm 2 when \(i=2\) is \(-2(2)+6 = 2)\nTerm 3 when \(i=3\) is \(-2(3)+6 = 0)

Step 2: Identify the last term

The last term (fourth term in this case) of the arithmetic sequence can be found by substituting \(i=4\) into the equation \(-2i+6\). This will give the term as follows:\nLast term (Term 4) when \(i=4\) is \(-2(4)+6 = -2)

Step 3: Apply the formula of the sum of an arithmetic sequence

The formula for the sum \(S\) of an arithmetic sequence is given by \(S=n/2*(a_1+a_n)\) where \(n\) is the number of terms, \(a_1\) is the first term and \(a_n\) is the last term. Substituting the appropriate values into the equation gives:\nSum = \(4/2 * (4 + (-2)) = 2 * 2 = 4\)

Key Concepts

sum of arithmetic sequence

Let's talk about the sum of an arithmetic sequence—a fancy way of saying you're adding up a list of numbers that have a regular pattern. In simpler terms, in an arithmetic sequence, you consistently add or subtract the same number, called the "common difference," to get to the next number.
For example, if you start with 4, and then each time subtract 2, you generate the sequence 4, 2, 0, -2, and so on.

• The formula to find the sum of an arithmetic sequence is: \[ S = \frac{n}{2} \times (a_1 + a_n) \ \]
• Here, \(S\) is the sum of the sequence.
• \(n\) is the total number of terms you're adding together.
• \(a_1\) is your first term, while \(a_n\) is the last term in the list.

Using this formula is super handy because you don't need to add up each number manually! You can find the sum in a flash, even with lots of terms.

first term of sequence

Finding the first term of an arithmetic sequence is pretty straightforward. You look at the sequence's starting point, the first number listed.
In the exercise, your sequence is represented by the expression \(-2i + 6\). To find the first term, replace \(i\) with 1.

• Let's plug \(i = 1\) into \(-2i + 6\).
• The calculation is \(-2(1) + 6 = 4\).

So, 4 is your first term here. Not too bad, right? Once you've got it, you can use it in the rest of your calculations, like finding the sum of the sequence.

last term of sequence

Locating the last term in an arithmetic sequence involves a similar process to finding the first term. You substitute the highest value of \(i\)—which corresponds to the position of the last term—into the sequence's expression.
In this exercise, that means using \(i = 4\), since the sequence stops after four terms.

• So, using \(-2i + 6\), substitute \(i = 4\).
• Do the math: \(-2(4) + 6 = -2\).

That's it! Your last term is \(-2\). Knowing both the first and last terms is crucial for calculating the sum of the sequence quickly.

number of terms in sequence

The number of terms in an arithmetic sequence simply refers to how many numbers you're working with. It's like counting the items in a list.
Commonly, this information is either given or can be deduced from the context. In our example, the sequence was written with a sum notation implying it starts at \(i = 1\) and ends at \(i = 4\).

• This tells us there are 4 terms.
• Every position \(i = 1, 2, 3, 4\) indicates another term.

With this number, you can go ahead to employ useful formulas, like the one to find the sum of the sequence. Knowing the number of terms helps tackle problems more swiftly and accurately.