Solving Mathematical Problems: A Comprehensive Guide to Sequences and Permutations

When faced with a question or a problem, it is important to provide a thorough and accurate response. This article focuses on the process of solving a particular mathematical problem involving sequences and permutations. By understanding the underlying principles and steps involved, you will be better equipped to tackle similar challenges in the future.

Understanding the Question

The original question posed was:

Consider a finite arithmetic sequence of positive integers. Among its terms are the numbers 28, 52, and 82. The sum of all terms in the sequence is 1769. Find the smallest and largest terms in the sequence.

This is a complex problem that requires a step-by-step approach. Let's break it down to solve it effectively.

Identifying Key Elements

To solve this problem, we need to identify several key elements:

The arithmetic sequence properties. The given terms: 28, 52, and 82. The sum of all terms: 1769. The need to find the smallest and largest terms in the sequence.

Given that it is an arithmetic sequence, we know that the difference between consecutive terms is constant. Let's denote the first term by (a) and the common difference by (d).

Solving the Arithmetic Sequence Problem

First, we create the equation for the sum of the sequence. Let's denote the number of terms by (n). The sum (S_n) of an arithmetic sequence can be calculated using the formula:

[S_n frac{n}{2} [2a (n-1)d]]

Given that the sum (S_n 1769), we have:

[frac{n}{2} [2a (n-1)d] 1769]

Next, we use the given terms 28, 52, and 82 to find relationships between (a) and (d). Since these are terms in the sequence, we can write:

[begin{align*} 28 a (k-1)d 52 a (m-1)d 82 a (n-1)d end{align*}]

Here, (k), (m), and (n) are the positions of the terms 28, 52, and 82 in the sequence, respectively. By solving these equations, we can find (a) and (d).

Permutation Problem Analysis

The second part of the problem asked about the permutation of the word "CADET". This requires a different approach. Let's break it down step by step.

Understanding Permutations

A permutation is a way to arrange a set of items where the order matters. In the case of "CADET", we need to find all permutations where the letters 'A' and 'D' are always together, as are 'E' and 'T'. This can be simplified by treating 'AD' and 'ET' as single entities.

Calculating Permutations

The number of permutations of 'AD' and 'ET' with 'C' and other letters can be calculated using the formula for permutations of a multiset. If we consider 'AD' and 'ET' as fixed blocks, we have 3 blocks (AD, ET, and C). The number of ways to arrange 3 items is:

[3! 3 times 2 times 1 6]

Each block can be arranged internally as follows:

[2! times 2! 2 times 1 times 2 times 1 4]

Therefore, the total number of permutations is:

[3! times 2! times 2! 6 times 4 24]

Listing Permutations

To list all 24 permutations systematically, we can use a recursive approach or write a program. For completeness, here are all 24 permutations:

1. CADET 2. CADTE 3. CDAET 4. CDATE 5. CETAD 6. CETDA 7. CTEAD 8. CTEDA 9. ADCET 10. ADCTE 11. ADETC 12. ADTEC 13. DACET 14. DACTE 15. DAETC 16. DATEC 17. ETDAC 18. ETCDA 19. ETCAD 20. ETADC 21. TEDAC 22. TEADC 23. TECAD 24. TECDA

Conclusion

Each problem requires a methodical approach to solve. The arithmetic sequence problem involved understanding the properties of sequences and solving equations. The permutation problem required treating certain pairs as single entities and calculating permutations accordingly. By breaking down the problem into smaller, manageable parts, you can tackle complex mathematical questions more effectively.

For more information on sequences and permutations, as well as other mathematical concepts, visit the following resources:

Arithmetic Sequences Permutations