Problem 22 - Algebra 2

Question

Find the missing term of each arithmetic sequence. \(14, \square, 28, \ldots\)

Step-by-Step Solution

Answer

The missing term in the arithmetic sequence is 21.

Step 1: Calculate the difference

First, find the difference between the two given numbers in the sequence. In this case, the given numbers are 14 and 28. The difference is \(28 - 14 = 14\). This difference, divided by the number of steps to get from 14 to 28, gives the common difference in our arithmetic sequence.

Step 2: Divide the difference

Now, divide the difference calculated in the previous step by the number of steps between the given numbers. Here, there is one missing term, so there are 2 steps (from 14 to the missing term and from the missing term to 28). Hence, the common difference is \(14 ÷ 2 = 7\).

Step 3: Find the missing term

Finally, add the common difference to the preceding term to find the missing term. So, the missing term is \(14 + 7 = 21\). Therefore, the arithmetic sequence is \(14, 21, 28, \ldots\)

Key Concepts

Common Difference

The common difference in an arithmetic sequence is the consistent interval between consecutive terms. It is what keeps the sequence growing or shrinking steadily. Identifying this common difference is the first crucial step in solving problems related to arithmetic sequences.
In our example sequence of 14, _, 28, you need to calculate how far apart the numbers are. You do this by subtracting the first known term from the subsequent known term.

• Given numbers in the sequence: 14 and 28.
• Calculate difference: \(28 - 14 = 14\).

This number is the total difference needed to traverse through the entire sequence span we have data for. To find out how much the sequence increments or decrements per step, divide by the number of gaps between terms. With one missing term, we divide by 2.

• Common difference calculation: \(14 \div 2 = 7\).

Now you have determined that each term is spaced by 7.

Missing Term in Sequence

Finding the missing term in an arithmetic sequence requires understanding the common difference. Once you have determined the common difference, locating the missing terms becomes straightforward.
In our sequence, 14, \( \square \), 28, once we know that each term increases by 7, it's easy to find the missing term by adding the common difference to the preceding term, which in this case is the number 14.

• Preceding term: 14.
• Common difference: 7.
• Missing term calculation: \(14 + 7 = 21\).

This calculation provides us with the missing number, which fills the gap accurately in our sequence. The sequence now reads 14, 21, 28. Each progression follows the calculated step of 7, confirming the correct solution.

Step-by-Step Solution

Solving arithmetic sequences can be easier if done in a clear, orderly fashion. By proceeding one step at a time, you ensure no errors are made. Let's break down the solution to the sequence 14, \(\square\), 28.
**Step 1: Calculate the Total Difference**

• Identify known terms: 14 and 28.
• Find total difference: \(28 - 14 = 14\).

**Step 2: Determine the Common Difference**

• Note the number of steps (spaces) between terms: 2 (14 to the missing term, then missing term to 28).
• Common difference: Total difference \(\div\) Number of steps = \(14 \div 2 = 7\).

**Step 3: Find the Missing Term**

• Add common difference to earlier number in sequence: \(14 + 7 = 21\).
• Complete sequence: 14, 21, 28.

These steps structure your problem-solving process, allowing you to logically determine missing elements in any arithmetic progression with confidence.