Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the sum of the finite arithmetic sequence. $$-2+(-5)+(-8)+(-11)+(-14)+(-17)$$

Step-by-Step Solution

Answer

-57

Step 1: Identify the First Term, Common Difference, and Number of Terms

The first term (a) in the arithmetic sequence is -2. The common difference (d), which can be found by subtracting any term from the term before it, is -3. The number of terms (n) can be determined by counting all the terms in the sequence, which is 6 in this case.

Step 2: Identify the Last Term

The last term (l) in the sequence is -17.

Step 3: Use the Formula for the Sum of Arithmetic Sequence

Now, plug in the known values into the formula for the sum of an arithmetic sequence, i.e., \(S_n = \frac{n}{2} (a + l)\). Substituting these values will give \(S_6 = \frac{6}{2} (-2 + (-17)) = 3 * -19 = -57\).

Key Concepts

Finite Sequence

A finite sequence is a sequence of numbers that contains a specific number of elements. In simpler terms, it's a list of numbers with a definite beginning and end. For example, in the sequence you've been given,
\(-2, -5, -8, -11, -14, -17\), there are exactly six elements.

Here are some important characteristics of a finite sequence:

• Definite Length: It always consists of a fixed number of terms.
• Specific Order: The order of the terms is crucial, altering the sequence disrupts its pattern.
• Clear End: Unlike infinite sequences, finite ones don't "go on forever." They end after a certain term.

Understanding finite sequences is essential, especially when calculating sums or analyzing the progression these numbers follow.

Common Difference

The term "common difference" is fundamental to understanding arithmetic sequences. It is the constant amount that each term in the sequence changes by.

You can find the common difference by subtracting any term from the term that follows it. In the given sequence, \(-2, -5, -8, -11, -14, -17\), each term is decreasing by 3:

• \(-5 - (-2) = -3\)
• \(-8 - (-5) = -3\)
• Continues for subsequent terms...

The common difference helps determine the position of any term in the sequence, allowing you to understand its pattern better.
This value is crucial when calculating the sum of the sequence.

Sum of Sequence

The sum of a sequence, especially a finite arithmetic one, is the total of all its terms added together. Calculating the sum is a common task that often requires applying formulas for accuracy and speed.

In arithmetic sequences, finding the sum can be simplified by using specific methods due to their predictable patterns. By understanding how each term is related through the common difference, it's possible to efficiently calculate the sum of extensive sequences without adding each term individually one by one.

The sum of the sequence is crucial in various fields of mathematics and real-world applications, making it important to learn how to compute it correctly.

Arithmetic Series Formula

To sum up an arithmetic sequence, you can use the arithmetic series formula. This formula is particularly useful for sequences with a large number of terms or complex numbers.

The formula for the sum \(S_n\) of an arithmetic sequence is:\[S_n = \frac{n}{2} (a + l)\]Where:

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

This formula effectively captures the pattern of the sequence by taking both the first and last terms into consideration, added then averaged over the number of terms.
For the sequence \(-2, -5, -8, -11, -14, -17\), applying the formula yields: \[S_6 = \frac{6}{2} \times (-2 + (-17)) = 3 \times -19 = -57\]This result reflects the total sum of the numbers in the sequence.