Proving Inequality: a and b Sequences, Sum and Product Relations

In summary, we use the given hint to prove that the sum of the products of two decreasing sequences is less than or equal to the product of their sums. By expanding the expression and simplifying, we arrive at the final inequality.
  • #1
cloud18
8
0
If a1 >= a2 >= ... >= a_n and b1 >= b2 >= ... >= b_n, prove that:

( Sum(a_k from k = 1 to n) )*( Sum(b_k from k = 1 to n) ) <= n*Sum(a_k*b_k
from k = 1 to n).

Hint: Sum( (a_k - a_j)*(b_k - b_j) s.t. 1 <= j <= k <= n ) >= 0

This is what I have so far:

0 <= Sum( (a_k - a_j)*(b_k - b_j) s.t. 1 <= j <= k <= n )

0 <= Sum( a_k*b_k - a_k*b_j - a_j*b_k + a_j*b_j s.t. 1 <= j <= k <= n )

0 <= Sum( a_k*b_k from k = 1 to n ) - Sum( a_k*b_j s.t. 1 <= j <= k <=
n ) - Sum( a_j*b_k s.t. 1 <= j <= k <= n ) + Sum( a_j*b_j from j = 1 to
n )

0 <= 2*Sum( a_k*b_k from k = 1 to n ) - Sum( a_k*b_j s.t. 1 <= j <= k <=
n ) - Sum( a_j*b_k s.t. 1 <= j <= k <= n )

I hope that is correct thus far, but do not know what to do next.
 
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  • #2
Answer:0 <= 2*Sum( a_k*b_k from k = 1 to n ) - Sum( a_k*b_j s.t. 1 <= j <= k <= n ) - Sum( a_j*b_k s.t. 1 <= j <= k <= n )0 <= 2*Sum( a_k*b_k from k = 1 to n ) - Sum( (a_k + a_j)*b_j s.t. 1 <= j <= k <= n )0 <= 2*Sum( a_k*b_k from k = 1 to n ) - n*Sum( a_j*b_j s.t. 1 <= j <= n )2*Sum( a_k*b_k from k = 1 to n ) >= n*Sum( a_j*b_j s.t. 1 <= j <= n )Sum(a_k from k = 1 to n) * Sum(b_k from k = 1 to n) = Sum( a_k*b_k from k = 1 to n ) >= n*Sum( a_j*b_j s.t. 1 <= j <= n )Sum(a_k from k = 1 to n) * Sum(b_k from k = 1 to n) <= n*Sum(a_k*b_k from k = 1 to n).
 

What is the purpose of proving inequality in a and b sequences?

The purpose of proving inequality in a and b sequences is to determine the relationship between two different sequences and to establish whether one sequence is always larger or smaller than the other. This can help in understanding the behavior of the sequences and making predictions about their values.

What are the common methods used to prove inequality in sequences?

There are several methods that can be used to prove inequality in sequences, including mathematical induction, direct proof, and contradiction. These methods involve using logical reasoning and mathematical principles to establish the relationship between the sequences.

How does proving inequality in sequences relate to sum and product relations?

Proving inequality in sequences often involves using the sums and products of the terms in the sequences. These sums and products can provide valuable information about the behavior of the sequences and can help in proving that one sequence is always greater or less than another.

What are some real-life applications of proving inequality in sequences?

Proving inequality in sequences has many real-life applications, such as in economics, where it can be used to analyze the growth or decline of certain values over time. It can also be applied in physics and engineering to understand the behavior of systems and make predictions about their future performance.

What are some potential challenges in proving inequality in sequences?

One challenge in proving inequality in sequences is identifying the appropriate method to use, as different sequences may require different approaches. Additionally, the complexity of the sequences and the lack of a clear pattern can make it difficult to establish a relationship between them. It is also important to consider the assumptions and limitations of the proof and account for any potential errors or exceptions.

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