Direction: Find the wrong term in the series given below.

1, 27, 83, 157, 261, 391

To identify the wrong term in the series: 1, 27, 83, 157, 261, 391, let's first analyze the pattern of differences between consecutive terms.

1. **Difference between the 2nd and 1st term:**
   \[
   27 - 1 = 26
   \]

2. **Difference between the 3rd and 2nd term:**
   \[
   83 - 27 = 56
   \]

3. **Difference between the 4th and 3rd term:**
   \[
   157 - 83 = 74
   \]

4. **Difference between the 5th and 4th term:**
   \[
   261 - 157 = 104
   \]

5. **Difference between the 6th and 5th term:**
   \[
   391 - 261 = 130
   \]

The differences between consecutive terms are:
- 26
- 56
- 74
- 104
- 130

Let's check if these differences follow any consistent pattern. Notice that if we compute the second differences (the differences between consecutive first differences):

1. **Second difference between 56 and 26:**
   \[
   56 - 26 = 30
   \]

2. **Second difference between 74 and 56:**
   \[
   74 - 56 = 18
   \]

3. **Second difference between 104 and 74:**
   \[
   104 - 74 = 30
   \]

4. **Second difference between 130 and 104:**
   \[
   130 - 104 = 26
   \]

These second differences should ideally form a consistent pattern, but they do not in this case. This inconsistency suggests there might be a mistake in the sequence.

Let's hypothesize the differences should follow a consistent pattern, and look for the most likely error:

- From the second differences, it's evident that the difference 18 is the anomaly since the pattern 30, 30, and 26 is somewhat more consistent. This discrepancy indicates that the third difference of 74 should instead have been part of a regular series. 
   
   Checking the sequence around the error, we find:
   \[
   \text{Corrected } 74 = 56 + 30 = 86
   \]

   Thus, the corrected series should be:

   \[
   1, 27, 83, 157, 261, 391
   \]

The wrong term is the **3rd term: 83**. It should have been **86** instead.

Would you like a detailed explanation on this process? If not, here are some related questions to deepen your understanding:

1. How do you find patterns in a series?
2. What are second differences, and how do they help in identifying errors in sequences?
3. How can you determine if a sequence follows an arithmetic or geometric pattern?
4. Can you provide an example where the differences between terms form a quadratic sequence?
5. What other methods can be used to find errors in numerical sequences?

**Tip:** Always check for consistent patterns in the first and second differences when analyzing sequences to identify potential errors.

Explore how to identify errors in numerical sequences using the series 1, 27, 83, 157, 261, 391 as an example. Learn about pattern recognition and second differences to pinpoint inconsistencies. Suitable for high school students and above.

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