Identifying the Missing Number in a Series Using Arithmetic Sequences

Are you stumped by series and sequence problems, particularly those involving finding the missing number? Let's explore one such problem: 133, 154, 175, X, 217, 238. This article will walk you through various methods to solve such problems using the properties of arithmetic sequences. By the end, you'll be able to recognize and solve similar series with ease.

Understanding Arithmetic Sequences

In an arithmetic sequence, each term is obtained by adding the same constant value to the previous term. This constant value is known as the common difference. In our series, the common difference is consistently 21:

This consistency allows us to apply our knowledge of arithmetic sequences to find missing numbers.

Method 1: Direct Addition

The simplest method involves directly adding the common difference to the preceding term to find the following term. Since the common difference is 21, and the term before X is 175, we can easily find X by adding 21 to 175:

175 21 196

Method 2: Algebraic Representation

Another approach is to use an algebraic equation to represent the sequence. If we assume the missing number X, we can set up an equation based on the constant difference:

X - 175 217 - X

By solving this equation:

2X 175 217

2X 392

X 196

Thus, the missing number is 196, which fits the pattern of the series.

Verification

To verify our solution, we can add 21 to 196 and check if we get the next number in the series:

196 21 217

Indeed, this confirms that 196 is the correct missing number.

General Term of the Sequence

For those interested in a more generalized approach, we can define the general term fn of the sequence as:

fn 21n 129

To find the third term (n 3), we substitute:

f3 21 × 3 129 63 129 192

This confirms our earlier solution and provides a formula to find any term in the series without manually adding the differences.

By understanding and applying these methods, you can solve similar series of numbers with confidence. Remember, practice is the key to mastery. Happy problem-solving!