Initial Approach using Pigeonhole Principle

Oct 13, 2025 by recoveryshake.com 47 views

Show that there is a block of $r$ successive $a_i$ 's $(r \ge 1)$ whose sum is $95$

This problem is taken from the book "An Excursion in Mathematics" and involves sequences and series, combinatorics, and the pigeonhole principle. The question asks us to prove that there exists a block of $r$ successive terms in a sequence $a_i$ whose sum is equal to $95$ . In this article, we will explore different approaches to solving this problem and provide a step-by-step solution.

Understanding the Problem

The problem statement is quite straightforward. We are given a sequence of numbers $a_i$ and we need to show that there exists a block of $r$ successive terms in this sequence whose sum is equal to $95$ . The value of $r$ is not specified, but it is mentioned that $r \ge 1$ . This means that we need to find at least one block of consecutive terms that adds up to $95$ .

Initial Approach using Pigeonhole Principle

The initial approach to solving this problem is to use the pigeonhole principle. The pigeonhole principle states that if $n$ items are put into $m$ containers, with $n > m$ , then at least one container must contain more than one item. In this case, we can consider the sequence $a_i$ as the items and the possible sums of $r$ consecutive terms as the containers.

However, upon closer inspection, we realize that this approach may not be the most effective way to solve the problem. The pigeonhole principle is typically used to prove the existence of at least one item in a container, but in this case, we need to prove the existence of a block of consecutive terms whose sum is equal to $95$ .

Alternative Approach

A more effective approach to solving this problem is to use a combination of mathematical reasoning and logical thinking. Let's consider the sequence $a_i$ as a list of numbers, and let's assume that the sum of the first $r$ terms is less than $95$ . In this case, we can add the next term to the sequence and see what happens.

If the sum of the first $r$ terms is less than $95$ , then the sum of the first $r+1$ terms must be greater than or equal to $95$ . This is because we have added a positive number to the sum, which must increase the sum by at least $1$ .

Now, let's consider the sum of the first $r+1$ terms. If this sum is greater than or equal to $95$ , then we have found a block of $r+1$ consecutive terms whose sum is equal to $95$ or greater. This is exactly what we need to prove.

Proof by Mathematical Induction

To prove that there exists a block of $r$ successive terms whose sum is equal to $95$ , we can use mathematical induction. The base case is when $r=1$ , in which case the sum of the first term is equal to $95$ . This is trivially true.

For the inductive step, we assume that the statement is true for some $k \ge 1$ . That is, we assume that there exists a block of $k$ successive terms whose sum is equal to $95$ . We need to show that the statement is true for $k+1$ .

Let's consider the sum of the first $k+1$ terms. If this sum is greater than or equal to $95$ , then we have found a block of $k+1$ consecutive terms whose sum is equal to $95$ or greater. This is exactly what we need to prove.

On the other hand, if the sum of the first $k+1$ terms is less than $95$ , then we can add the next term to the sequence and see what happens. As before, the sum of the first $k+2$ terms must be greater than or equal to $95$ . This is because we have added a positive number to the sum, which must increase the sum by at least $1$ .

Therefore, by mathematical induction, we have shown that there exists a block of $r$ successive terms whose sum is equal to $95$ for all $r \ge 1$ .

In this article, we have explored different approaches to solving the problem of showing that there exists a block of $r$ successive terms in a sequence $a_i$ whose sum is equal to $95$ . We have used a combination of mathematical reasoning and logical thinking to prove the statement by mathematical induction. The proof is straightforward and easy to follow, and it provides a clear understanding of the problem and its solution.

The problem of showing that there exists a block of $r$ successive terms in a sequence $a_i$ whose sum is equal to $95$ is a classic example of a problem that can be solved using mathematical induction. The proof is elegant and easy to follow, and it provides a clear understanding of the problem and its solution. We hope that this article has provided a useful insight into the problem and its solution, and we encourage readers to try solving the problem on their own.

"An Excursion in Mathematics" (book)
Pigeonhole Principle (mathematical concept)
Mathematical Induction (mathematical concept)

Sequences and series
Combinatorics
Pigeonhole principle
Mathematical induction
Block of consecutive terms
Sum of terms
Mathematical reasoning
Logical thinking
Q&A: Show that there is a block of $r$ successive $a_i$ 's $(r \ge 1)$ whose sum is $95$ ===========================================================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions related to the problem of showing that there exists a block of $r$ successive terms in a sequence $a_i$ whose sum is equal to $95$ .

Q: What is the main idea behind the solution to this problem?

A: The main idea behind the solution to this problem is to use mathematical induction to prove that there exists a block of $r$ successive terms in a sequence $a_i$ whose sum is equal to $95$ for all $r \ge 1$ .

Q: Why do we need to use mathematical induction to solve this problem?

A: We need to use mathematical induction to solve this problem because it allows us to prove the statement for all $r \ge 1$ by considering the base case and the inductive step.

Q: What is the base case in the mathematical induction proof?

A: The base case in the mathematical induction proof is when $r=1$ , in which case the sum of the first term is equal to $95$ . This is trivially true.

Q: What is the inductive step in the mathematical induction proof?

A: The inductive step in the mathematical induction proof is to assume that the statement is true for some $k \ge 1$ , and then to show that the statement is true for $k+1$ .

Q: How do we prove the inductive step in the mathematical induction proof?

A: We prove the inductive step in the mathematical induction proof by considering the sum of the first $k+1$ terms. If this sum is greater than or equal to $95$ , then we have found a block of $k+1$ consecutive terms whose sum is equal to $95$ or greater. If the sum of the first $k+1$ terms is less than $95$ , then we can add the next term to the sequence and see what happens.

Q: What is the significance of the pigeonhole principle in this problem?

A: The pigeonhole principle is not directly used in the solution to this problem, but it can be used to provide an alternative approach to solving the problem.

Q: Can you provide an example of a sequence $a_i$ for which there exists a block of $r$ successive terms whose sum is equal to $95$ ?

A: Yes, here is an example of a sequence $a_i$ for which there exists a block of $r$ successive terms whose sum is equal to $95$ :

$a_1 = 1, a_2 = 2, a_3 = 3, a_4 = 4, a_5 = 5, a_6 = 6, a_7 = 7, a_8 = 8, a_9 = 9, a_{10} = 10$

In this sequence, the sum of the first $10$ terms is equal to $55$ , and the sum of the next $10$ terms is equal to $40$ . Therefore, there exists a block of $20$ consecutive terms whose sum is equal to $95$ .

Q: Can you provide a general formula for the sum of a block of $r$ successive terms in a sequence $a_i$ ?

A: Yes, here is a general formula for the sum of a block of $r$ successive terms in a sequence $a_i$ :

$S_r = a_1 + a_2 + a_3 + \cdots + a_r$

where $S_r$ is the sum of the first $r$ terms, and $a_i$ is the $i$ th term in the sequence.

Q: Can you provide a general formula for the sum of a block of $r$ successive terms in a sequence $a_i$ that starts at the $k$ th term?

A: Yes, here is a general formula for the sum of a block of $r$ successive terms in a sequence $a_i$ that starts at the $k$ th term:

$S_{r,k} = a_k + a_{k+1} + a_{k+2} + \cdots + a_{k+r}$

where $S_{r,k}$ is the sum of the $r$ terms starting at the $k$ th term, and $a_i$ is the $i$ th term in the sequence.

In this article, we have answered some of the most frequently asked questions related to the problem of showing that there exists a block of $r$ successive terms in a sequence $a_i$ whose sum is equal to $95$ . We hope that this article has provided a useful insight into the problem and its solution, and we encourage readers to try solving the problem on their own.

"An Excursion in Mathematics" (book)
Pigeonhole Principle (mathematical concept)
Mathematical Induction (mathematical concept)

Sequences and series
Combinatorics
Pigeonhole principle
Mathematical induction
Block of consecutive terms
Sum of terms
Mathematical reasoning
Logical thinking