Problem 79 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I was able to find the sum of the first 50 terms of an arithmetic sequence even though I did not identify every term.

Step-by-Step Solution

Answer

The statement makes sense because the sum of an arithmetic sequence can be calculated using the formula for the sum of an arithmetic sequence without necessarily knowing each individual term. All that's needed are the first term, the common difference, and the number of terms.

Step 1: Understand the properties of an arithmetic sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference. Examples of arithmetic sequences include 2, 4, 6, 8, 10, ..., with a common difference of 2, or 100, 90, 80, 70, ..., with a common difference of -10.

Step 2: Review the formula for the sum of an arithmetic sequence

The formula for the sum of the first n terms (S) of an arithmetic sequence is given by: S_n = n/2 [2a + (n-1)d] where: n is the number of terms, a is the first term, and d is the common difference.

Step 3: Apply the properties and formula to the given statement

In the given statement, an arithmetic sequence of 50 terms is mentioned. Since the sum of an arithmetic sequence can be calculated using the formula mentioned in step 2 without necessarily knowing each individual term, the statement 'I was able to find the sum of the first 50 terms of an arithmetic sequence even though I did not identify every term.' makes sense.

Key Concepts

Arithmetic Sequence Properties

An arithmetic sequence is a series of numbers in which each term is derived by adding a constant value to the previous term. This constant value is known as the common difference, symbolically represented by the letter d. For a sequence to qualify as arithmetic, this common difference must be consistent throughout.

For instance, in the sequence 3, 7, 11, 15..., the common difference is 4 since each subsequent term is 4 more than the previous term. Understanding these properties allows learners to analyze and make predictions about arithmetic sequences. So, even without every individual term, one can still apply the general properties to calculate certain attributes of the sequence such as its sum, which demonstrates how powerful a grasp of the fundamental properties can be in arithmetic sequences.

Arithmetic Sequence Formula

When working with arithmetic sequences, a key tool is the sum formula. It enables the calculation of the sum of a sequence without the need to sum each individual term.

The arithmetic series sum formula is expressed as:
\[ S_n = \frac{n}{2} [2a + (n - 1)d] \]
where:

• n denotes the number of terms,
• a represents the first term in the sequence,
• and d is the sequence's common difference.

Imagine needing to sum the first 100 terms of an arithmetic sequence. Rather than adding all one hundred terms, you can simply apply this formula, input the necessary values, and swiftly determine the sum. This method is not only accurate but also greatly reduces the complexity and time required for such calculations.

Common Difference in Arithmetic Sequence

The common difference plays a pivotal role in defining an arithmetic sequence. It is the fixed amount added to each term to get the next term and is denoted as d. Think of it as the engine that drives the sequence forward, maintaining equal distances between each term. The common difference can be positive, negative, or even zero.

To find the common difference, take any term in the sequence (after the first) and subtract the term before it:
\[ d = a_{n} - a_{n-1} \]
For example, in the sequence 5, 8, 11, 14..., the common difference is 3 because each term increases by 3. In the reverse, if you encounter a sequence like 20, 15, 10, 5..., the common difference is -5. Understanding and identifying the common difference is essential when applying the arithmetic sequence formula to find the sum or other characteristics of the sequence.