Arithmetic string equal to the geometric string, count x

In summary: This means that the product on the left-hand side is greater than the sum on the right-hand side, which contradicts the given condition that they are equal. Therefore, our assumption that there exists a larger number in the set is false, and 1 is indeed the maximum possible value for the largest number in the set. In summary, the maximum possible value of the largest number in this scenario is 25, which can be achieved by setting all 25 numbers to 1. This can be proven by showing that any other number greater than 1 in the set would lead to a contradiction.
  • #1
ghostfirefox
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Let x1, ..., x25 be such positive integers that x1⋅x2⋅ ... ⋅x25 = x1 + x2 + ... + x25. What is the maximum possible value of the largest of numbers x1, x2, ..., x25?
 
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The maximum possible value of the largest number in this scenario is 25. This can be achieved by setting all 25 numbers to 1, as 1+1+...+1 = 25 and 1*1*...*1 = 25.

To prove that this is the maximum possible value, let's assume that there exists a larger number, say n, in the set of x1, x2, ..., x25. This would mean that n is greater than 1, since we have already established that 1 is the maximum value.

Since n is a positive integer, it can be written as a sum of positive integers, let's say n = a + b, where a and b are positive integers.

Now, let's consider the product of all 25 numbers: x1 * x2 * ... * x25. From the given condition, we know that this product is equal to the sum of all 25 numbers: x1 + x2 + ... + x25.

Since n is one of the numbers in this product and sum, we can replace it with a + b. This gives us:

x1 * x2 * ... * x25 = x1 + x2 + ... + x25

Replacing n with a + b, we get:

x1 * x2 * ... * x25 = x1 + x2 + ... + x25 - n + n

Since n = a + b, we can rewrite this as:

x1 * x2 * ... * x25 = x1 + x2 + ... + x25 - (a + b) + (a + b)

Now, we can rearrange the terms in the sum on the right-hand side to get:

x1 * x2 * ... * x25 = (x1 - a) + (x2 - b) + ... + (x25 - b) + (a + b)

We know that n = a + b and n is one of the numbers in the sum on the right-hand side. Therefore, we can replace n with a + b to get:

x1 * x2 * ... * x25 = (x1 - a) + (x2 - b) + ... + (x25 - b) + n

Since n is greater than 1, we know that at least one of the numbers in the sum on
 

Related to Arithmetic string equal to the geometric string, count x

1. What does the phrase "Arithmetic string equal to the geometric string" mean?

The phrase "Arithmetic string equal to the geometric string" refers to a sequence of numbers where each term is obtained by adding a constant number to the previous term, and a geometric sequence where each term is obtained by multiplying the previous term by a constant number.

2. What is the purpose of counting x in this context?

Counting x in this context refers to determining the number of terms in the arithmetic sequence that are equal to the corresponding terms in the geometric sequence. This can help identify patterns and relationships between the two sequences.

3. How do you determine if an arithmetic string is equal to a geometric string?

To determine if an arithmetic string is equal to a geometric string, you can compare the terms of the two sequences. If each term in the arithmetic sequence is equal to the corresponding term in the geometric sequence, then the two strings are equal.

4. Can an arithmetic string ever be equal to a geometric string?

Yes, an arithmetic string can be equal to a geometric string. This can occur when the constant number in the arithmetic sequence is equal to the constant number in the geometric sequence, or when the two sequences have a common ratio.

5. How is this concept used in real-world applications?

The concept of an arithmetic string equal to a geometric string is used in a variety of fields, such as finance, physics, and computer science. In finance, it can be used to calculate compound interest. In physics, it can be used to model exponential growth or decay. In computer science, it can be used in algorithms and data structures.

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