Problem 51 - Discrete Mathematics with Applications

Question

Expand each. $$\sum_{1 \leq i

Step-by-Step Solution

Answer

The short answer to expand the given summation \(\sum_{1 \leq i

Step 1: Identify valid combinations of i and j

According to the given conditions, i and j must be among 1, 2, and 3, with i being less than j. Write down all the possible combinations that fulfill these conditions.

Step 2: Sum the elements for the identified combinations

For each valid combination of i and j found in step 1, sum the elements \(a_{i} + a_{j}\) and write them down.

Step 3: Add the sums to find the final result

Sum up the elements obtained in step 2 to get the final result of the summation. Now, let's perform these steps.

Step 1: Identify valid combinations of i and j

We have the following valid combinations of i and j that satisfy \(1 \leq i < j \leq 3\): (i, j) = (1, 2) \ (i, j) = (1, 3) \ (i, j) = (2, 3)

Step 2: Sum the elements for the identified combinations

For the combinations obtained in step 1, find the sum of the elements \(a_{i} + a_{j}\). Combination (1, 2): \(a_{1} + a_{2}\) Combination (1, 3): \(a_{1} + a_{3}\) Combination (2, 3): \(a_{2} + a_{3}\)

Step 3: Add the sums to find the final result

Finally, add the sums obtained in step 2 to get the final result of the summation: \(\sum_{1 \leq i

Key Concepts

Combinatorial Summation

Combinatorial summation involves finding the sum of elements based on specific combinations from a given set. It is about adding values according to some rules about how indices relate. For the problem given, we need to sum over all unordered pairs \((i, j)\) where the indices satisfy the condition \(1 \leq i < j \leq 3\). This means:

• We select any two different indices \(i\) and \(j\) from \(\{1, 2, 3\}\).
• We ensure that \(i\) is always less than \(j\) to prevent double-counting and maintain a proper order.

Combinatorial summation is helpful in scenarios where exact combinations of indices are calculated to yield meaningful results, like counting specific pairs or choosing subsets.

Indexing and Bounds in Summation

Indexing and bounds in summation define the limits and rules for valid index values. These bounds affect which terms are included in the summation. In our example, we have:

• Lower bound: \(i = 1\), which ensures that our index \(i\) starts from 1.
• Upper bound: \(j = 3\), which confines our index \(j\) to a maximum of 3.
• The condition \(i < j\) which is crucial for forming a proper, ordered pair.

Understanding these bounds helps correctly identify all possible terms needed for a combinatorial summation. Without respecting these limits, terms could potentially be repeated or omitted, leading to an incorrect summation outcome.

Series Expansion

Series expansion in discrete mathematics involves expressing a complex summation as the sum of simpler individual terms. For our problem, the goal is to expand:\[\sum_{1 \leq i

\((a_{1} + a_{2})\)

\((a_{1} + a_{3})\)

\((a_{2} + a_{3})\)

Each term results from a unique combination of indices, summed according to the specified rule. The series expansion thus simplifies the computation by breaking down a collective expression into manageable parts, proving particularly helpful for large numbers of terms, where explicit computation might otherwise be cumbersome.

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Problem 51

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Problem 52

Expand each. $$\sum_{1 \leq i \leq j

View solution

Key Concepts

Combinatorial Summation

Combinatorial summation involves finding the sum of elements based on specific combinations from a given set. It is about adding values according to some rules about how indices relate. For the problem given, we need to sum over all unordered pairs \((i, j)\) where the indices satisfy the condition \(1 \leq i < j \leq 3\). This means:

• We select any two different indices \(i\) and \(j\) from \(\{1, 2, 3\}\).
• We ensure that \(i\) is always less than \(j\) to prevent double-counting and maintain a proper order.

Combinatorial summation is helpful in scenarios where exact combinations of indices are calculated to yield meaningful results, like counting specific pairs or choosing subsets.

Indexing and Bounds in Summation

Indexing and bounds in summation define the limits and rules for valid index values. These bounds affect which terms are included in the summation. In our example, we have:

• Lower bound: \(i = 1\), which ensures that our index \(i\) starts from 1.
• Upper bound: \(j = 3\), which confines our index \(j\) to a maximum of 3.
• The condition \(i < j\) which is crucial for forming a proper, ordered pair.

Understanding these bounds helps correctly identify all possible terms needed for a combinatorial summation. Without respecting these limits, terms could potentially be repeated or omitted, leading to an incorrect summation outcome.

Series Expansion

Series expansion in discrete mathematics involves expressing a complex summation as the sum of simpler individual terms. For our problem, the goal is to expand:\[\sum_{1 \leq i

\((a_{1} + a_{2})\)

\((a_{1} + a_{3})\)

\((a_{2} + a_{3})\)

Each term results from a unique combination of indices, summed according to the specified rule. The series expansion thus simplifies the computation by breaking down a collective expression into manageable parts, proving particularly helpful for large numbers of terms, where explicit computation might otherwise be cumbersome.