Problem 71 - Algebra 1: Concepts and Skills

Question

List the next three numbers suggested by the sequence. (Skills Review pp. 781) $$ 1,3,5,7, ?, ?, ? $$

Step-by-Step Solution

Answer

The next three numbers in the sequence are 9, 11, 13.

Step 1: Identify the pattern

In this numerical sequence, the pattern is that each number is 2 more than the previous one. This can be confirmed by subtracting the previous number from the current number i.e \(3-1=2\), \(5-3=2\), \(7-5=2\). This difference is constant so it's an arithmetic sequence with a common difference of 2.

Step 2: Apply the pattern to find the next numbers

Knowing the common difference, the next number in the sequence will be: \(7 (last number) + 2 (common difference) = 9\). Repeat this process to find the next two numbers: \(9+2=11\), \(11+2=13\).

Key Concepts

Pattern Recognition

Pattern recognition is like a detective finding clues. Each number in a sequence gives a hint about the next number. Recognizing these clues requires looking at each number and understanding the relationship between them. In the given exercise, the sequence starts with 1, 3, 5, and 7. At first glance, these numbers might just look like a random list, but there's actually a hidden pattern.
By examining how the numbers change from one to the next, you can spot this pattern. When you subtract the first number from the second number (\(3 - 1 = 2\)), you find the difference is 2. Do the same for the next pairs (\(5 - 3 = 2\) and \(7 - 5 = 2\)), and you'll notice the pattern holds. This tells you that each number in the sequence is increased by 2.
Once you find a consistent pattern, it's like having a recipe. You can easily predict the following numbers without guessing. Recognizing patterns is useful because it helps simplify problems, making them easier to solve.

Common Difference

The common difference is the arithmetic sequence's best friend. In simple terms, it is the constant amount that you add to each number to get the next one. This uniformity is what makes it an arithmetic sequence. In the sequence given, the common difference is 2.

• Common difference = \(3 - 1 = 2\)
• Common difference = \(5 - 3 = 2\)
• Common difference = \(7 - 5 = 2\)

Knowing the common difference opens up the entire sequence for you. Once you have identified this difference, you can confidently calculate the numbers that follow.
Simply add this difference to the last known number to predict the next ones. It's like setting your bike on cruise control; you continually move forward at the same steady pace, no surprises. This understanding makes predicting and verifying each number in the sequence quite straightforward.

Numerical Sequence

A numerical sequence is like a train of numbers, where each carriage (number) holds a specific place, connected by a common mathematical rule. In the world of sequences, each number builds upon the previous one to create an organized line of values. With arithmetic sequences, each number is formed by adding the common difference to the previous number.
The sequence in this exercise begins with these numbers: 1, 3, 5, 7. Using the rule of our arithmetic sequence, with a common difference of 2, you can find the subsequent numbers by simply continuing the trend.

• Start with 7: 7 + 2 = 9
• Then add 2 to 9: 9 + 2 = 11
• Finally, add 2 to 11: 11 + 2 = 13

Thus, the sequence extends seamlessly. Understanding numerical sequences is crucial; they simplify complex problems into manageable steps, allowing you to address each number systematically and accurately. Recognizing how sequences work is essential in building foundations for more advanced math problems.