Problem 29 - Algebra 2

Question

Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms. $$ 25,50,75,100, \ldots $$

Step-by-Step Solution

Answer

The number sequence is an arithmetic sequence. The next two terms are 125 and 150.

Step 1: Identifying the Type of Sequence

To determine if the sequence is arithmetic, subtract the second term from the first term, the third term from the second term, and so on. In this sequence, 50-25 equals 25, 75-50 equals 25, and 100-75 equals 25, so the difference is constant. This indicates that this is an arithmetic sequence.

Step 2: Finding the Next Terms of the Sequence

Since this is an arithmetic sequence with a constant difference of 25, this difference can be added to the last given term, which is 100, to find the next term in the sequence. So, 100 + 25 equals 125. To find the following term, add the difference to the term just found. In this case, 125 + 25 equals 150.

Key Concepts

Sequence Identification

In mathematics, a sequence is a list of numbers arranged in a specific order. One of the most common types is the arithmetic sequence. To identify if a sequence is arithmetic, you need to check if there is a constant difference between consecutive terms. For the sequence 25, 50, 75, 100, you subtract one term from the next. If this difference is the same throughout the sequence, it's arithmetic. In our example:

• 50 - 25 = 25
• 75 - 50 = 25
• 100 - 75 = 25

Since each subtraction results in the same number, our sequence is indeed arithmetic. Recognizing this kind of pattern helps in solving a lot of problems related to sequences.

Common Difference

The common difference in an arithmetic sequence is the difference between any two consecutive terms. It's what separates one number from the next in the sequence. Knowing the common difference is crucial because it allows you to predict and continue the sequence. In our sequence, 25 is the common difference because:

• Every number increases by 25 from the one before it.

Simply put, if you know any term in the sequence, you can add or subtract the common difference to find the others. It's a key aspect of understanding arithmetic progressions since it defines the pattern of growth throughout.

Next Terms in Sequence

Once you identify a sequence as arithmetic and determine the common difference, you can easily find the next terms. In an arithmetic sequence like 25, 50, 75, 100, start with the last known term and add the common difference to predict future terms. If we begin with 100:

• Add 25 (the common difference) to get 125.
• Then, add another 25 to 125 to get 150.

So, the next two terms are 125 and 150. This methodical approach allows you to extend the sequence indefinitely, making it a powerful tool for solving sequence problems in math.