Problem 49 - Prealgebra

Question

Describe the sequence using words. $$10,21,32,43,54, \dots$$

Step-by-Step Solution

Answer

The sequence starts at 10 and increases by 11 each time.

Step 1: Identify the Pattern

Observe the sequence: 10, 21, 32, 43, 54. Notice that each term is larger than the previous one, indicating an arithmetic sequence.

Step 2: Determine the Common Difference

Subtract the first term from the second term: 21 - 10 = 11. Do the same for the next pair: 32 - 21 = 11. Continue checking a few more pairs to confirm. The common difference is 11.

Step 3: Describe the Sequence Rule

Using the first term and the common difference, describe the sequence: Start with 10, and add 11 to get the next term.

Step 4: Verify the Sequence

Confirm that adding 11 repeatedly generates the sequence: 10 + 11 = 21, 21 + 11 = 32, 32 + 11 = 43, and 43 + 11 = 54. This matches the given sequence.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a fundamental concept. It is the constant amount that separates consecutive terms. To find the common difference, subtract the first term from the second term in the sequence. For example, in the sequence provided: 10, 21, 32, 43, 54, the calculation is:

• Second term minus the first term: \(21 - 10 = 11\)
• Similarly, \(32 - 21 = 11\)

This consistent result, 11, is what we call the 'common difference.' Ensuring that this difference remains constant across all term pairs is crucial for confirming an arithmetic sequence. If each pair has the same difference, you can be confident it's an arithmetic sequence.

Sequence Rule

The sequence rule is what allows us to predict any term in the arithmetic sequence without guessing. Once you have the first term and know the common difference, you can define a simple rule. For the sequence 10, 21, 32, 43, 54, the first term is 10, and the common difference is 11.So, the sequence rule can be written generally as:\[ a_n = a_1 + (n-1) imes d \]Where:

• \(a_n\) is the \(n^{th}\) term
• \(a_1\) is the first term (10 in our example)
• \(d\) is the common difference (11 here)

Applying this formula helps you find any term. For instance, the third term is:\( a_3 = 10 + (3-1) \times 11 = 32 \). Using the rule ensures you can skip between terms effortlessly.

Number Patterns

Number patterns refer to sequences that follow a particular order. In arithmetic sequences, the number pattern is very straightforward. Here, each number is obtained by adding a constant called the 'common difference' to the previous one. Recognizing these patterns aids in predicting subsequent numbers in the sequence. Consider our sequence: 10, 21, 32, 43, 54. Not only does this sequence rise by 11 each time, but it also reveals a linear pattern if expressed graphically. Each point lies equidistant apart, projecting a straight line. To generalize:

• The pattern identifies predictable changes.
• Forms the base for many mathematical calculations.
• Offers a simplified approach to complex problems.

Identifying and utilizing these patterns can significantly enhance your problem-solving skills in arithmetic progressions.