Problem 56 - Algebra and Trigonometry

Question

Write the sum without using sigma notation. $$\sum_{k=6}^{9} k(k+3)$$

Step-by-Step Solution

Answer

The sum is 320.

Step 1: Analyze Sigma Notation

The sigma notation \( \sum_{k=6}^{9} k(k+3) \) represents the sum of the expression \( k(k+3) \) from \( k = 6 \) to \( k = 9 \). This means we need to substitute each integer from 6 to 9 into the expression \( k(k+3) \) and add the results.

Step 2: Substitute and Expand Terms

Replace \( k \) with each integer value between 6 and 9 in the expression \( k(k+3) \). Calculate each term individually:- For \( k = 6 \): \( 6(6+3) = 6 \times 9 = 54 \)- For \( k = 7 \): \( 7(7+3) = 7 \times 10 = 70 \)- For \( k = 8 \): \( 8(8+3) = 8 \times 11 = 88 \)- For \( k = 9 \): \( 9(9+3) = 9 \times 12 = 108 \)

Step 3: Add the Expanded Terms

Add up the expanded terms from the previous step:- First, add \( 54 + 70 \) to get \( 124 \).- Next, add \( 124 + 88 \) to get \( 212 \).- Finally, add \( 212 + 108 \) to get \( 320 \).

Key Concepts

Sigma Notation

Sigma notation is a concise and efficient way to represent the sum of a sequence of expressions. It uses the Greek letter sigma (\(\Sigma\)) to indicate the mathematical process of summation. The notation includes:

• An expression, which is a mathematical formula involving a variable.
• Limits of summation, which specify the starting and ending values for the variable.
• The variable of summation, usually denoted by a letter like \(k\).

In the exercise, sigma notation \(\sum_{k=6}^{9} k(k+3)\) tells us to replace \(k\) with each integer from 6 to 9 in the expression \(k(k+3)\). We then calculate these values and sum them up. This compact representation simplifies the way mathematicians and students approach problems requiring summation of sequential terms.

Arithmetic Series

An arithmetic series is the result of summing the terms of an arithmetic sequence, where each term is obtained by adding a constant difference to the previous term. Arithmetic sequences are defined by their first term and the common difference between consecutive terms.
For example, consider the sequence: 6, 7, 8, 9. Here, each number is 1 more than the previous number, making 1 the common difference. However, in this exercise, we don't have a classic arithmetic sequence because of the function \(k(k+3)\).
Nonetheless, the arithmetic idea helps us clearly order our tasks. When we compute terms like \(6(6+3)\), \(7(7+3)\), etc., the increasing value of \(k\) ensures we compute these terms in a consistent manner. After calculating each, adding them results in the arithmetic series sum, a typical step in discrete mathematics.

Summation

Summation involves the addition of a sequence of numbers; it’s a fundamental operation in mathematics. In the exercise, the task was to sum the values resulting from substituting integers 6 to 9 into the expression \(k(k+3)\).
To achieve this, the calculated terms were:\(54, 70, 88,\) and \(108\). These were found by:

• Substituting 6 into \(k(k+3)\) to get\(54\)
• Substituting 7 into \(k(k+3)\) to get \(70\)
• Substituting 8 into \(k(k+3)\) to get \(88\)
• Substituting 9 into \(k(k+3)\) to get \(108\)

After evaluating each term, they are summed sequentially: \(54 + 70 + 88 + 108 = 320\). The process of summation is crucial in converting a series of computations into a single cohesive result, demonstrating the power of breaking down and aggregating mathematical information.