Problem 46 - College Algebra

Question

For Exercises \(45-50,\) 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. $$\sum_{i=1}^{20}(6 i-4)$$

Step-by-Step Solution

Answer

The sum of the first 20 terms of the sequence is 1180.

Step 1: Find the first three terms

Substitute 1, 2, and 3 into the expression for i to find the first three terms of the sequence: \(T_1 = 6*1 - 4 = 2, T_2 = 6*2 - 4 = 8, T_3 = 6*3 - 4 = 14\).

Step 2: Find the last term

The formula for the arithmetic sequence is \(6i - 4\). Given that there are 20 terms, substitute 20 for i to find the last term: \(T_{20} = 6*20 - 4 =116\).

Step 3: Use the formula for sum of an arithmetic sequence

Plugging the obtained values into the formula for the sum of an arithmetic sequence, \(S = \frac{n}{2} * (a + l)\), and simplifying, we get \(S = \frac{20}{2} * (2 + 116) = 10 * 118 = 1180\).

Key Concepts

Sum of Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two successive members is a constant. This difference is known as the common difference. When you're asked to find the sum of an arithmetic sequence, you're essentially summing up all the terms in this sequence from the first term to the last.

For example, if you have 20 terms in your sequence, you need to add the first term, the second term, all the way through to the 20th term. Calculating each one individually would be tedious, but thankfully, there's a very convenient formula. This formula makes it much simpler to find the total sum without having to add up each number manually.

First Three Terms

To find the first three terms of an arithmetic sequence, you use the general expression for the sequence. This exercise provides the expression as \(6i-4\). By substituting different values for \(i\), typically starting from \(i = 1\), you can find the first few terms.

• For \(i = 1\), the first term \(T_1 = 6 \times 1 - 4 = 2\).
• For \(i = 2\), the second term \(T_2 = 6 \times 2 - 4 = 8\).
• For \(i = 3\), the third term \(T_3 = 6 \times 3 - 4 = 14\).

Knowing these initial terms helps you understand the pattern of the arithmetic sequence.

Last Term

The last term in an arithmetic sequence can be found using the same expression by substituting the last sequential position number. If you're looking at the 20th term, plug in \(i = 20\) into the formula \(6i-4\).

Doing this calculation, you get \(T_{20} = 6 \times 20 - 4 = 116\). The last term is vital because it's used in the formula to find the sum of the entire sequence. Identifying the last term helps in efficiently applying the sum formula.

Formula for Arithmetic Sequence Sum

The formula to find the sum \(S\) of an arithmetic sequence is \(S = \frac{n}{2}(a + l)\), where:

• \(n\) is the number of terms in the sequence.
• \(a\) is the first term.
• \(l\) is the last term.

To apply this formula, you simply substitute these values in. Given our example where \(n = 20\), \(a = 2\), and \(l = 116\), the sum becomes:
\[ S = \frac{20}{2}(2 + 116) = 10 \times 118 = 1180 \]
This formula significantly simplifies the process of finding the total sum of terms in an arithmetic sequence, saving both time and effort compared to manual addition.