Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

\(\sum_{k=3}^{6} \frac{k-5}{k-1}\) $$

Step-by-Step Solution

Answer

The sum is \(-\frac{14}{15}\).

Step 1: Understand the Problem

We need to evaluate a finite sum from \(k = 3\) to \(k = 6\) of the expression \(\frac{k-5}{k-1}\). This involves substituting each integer value of \(k\) into the expression and summing the results.

Step 2: Evaluate Expression for Each Term

Calculate \(\frac{k-5}{k-1}\) for each integer value of \(k\) from 3 to 6:- For \(k = 3\), \(\frac{3-5}{3-1} = \frac{-2}{2} = -1\).- For \(k = 4\), \(\frac{4-5}{4-1} = \frac{-1}{3}\).- For \(k = 5\), \(\frac{5-5}{5-1} = \frac{0}{4} = 0\).- For \(k = 6\), \(\frac{6-5}{6-1} = \frac{1}{5}\).

Step 3: Sum All the Values

Add the results from each term:\(-1 + \frac{-1}{3} + 0 + \frac{1}{5}\).

Step 4: Simplify the Sum

Find a common denominator to add the fractions:The least common denominator for 3 and 5 is 15.Convert and sum:\[-1 = \frac{-15}{15},\quad \frac{-1}{3} = \frac{-5}{15},\quad \frac{1}{5} = \frac{3}{15}\]Now compute:\[\frac{-15}{15} + \frac{-5}{15} + \frac{0}{15} + \frac{3}{15} = \frac{-17 + 3}{15} = \frac{-14}{15}\].

Step 5: Final Step: Conclusion

The value of the sum \( \sum_{k=3}^{6} \frac{k-5}{k-1} \) is \(-\frac{14}{15}\).

Key Concepts

Summation Notation

Summation notation is a mathematical shorthand used to denote the sum of a sequence of terms. It is commonly represented by the Greek letter sigma (\(\sum\)). This notation is particularly useful when dealing with long sums, allowing us to focus more on the calculation process rather than writing out each term individually. In our exercise, the summation notation is \(\sum_{k=3}^{6} \frac{k-5}{k-1}\), which instructs us to evaluate the expression \(\frac{k-5}{k-1}\) for each integer from 3 to 6, then to sum the results.

Understanding how to read summation notation is crucial:

• The variable below the sigma symbol (\(k\) in this case) is called the index of summation, which tells us what variable to change.
• The number beneath the sigma (3) is the starting point for \(k\).
• The number above the sigma (6) is the ending point for \(k\).

This sets our task as calculating the value of the series from \(k = 3\) to \(k = 6\). It's a powerful tool in calculus and algebra for expressing and calculating sums succinctly.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They are frequently encountered in algebra and calculus.

In our example, the expression \(\frac{k-5}{k-1}\) is a rational expression, where \(k-5\) is the polynomial in the numerator and \(k-1\) is the polynomial in the denominator. The key feature of a rational expression is that it can be undefined for certain values of the variables due to division by zero. However, in our given range (\(k = 3\) to \(k = 6\)), the denominator \(k-1\) is never zero.

To evaluate a rational expression:

• Substitute the variable with the given values or the values within a range.
• Perform the arithmetic operation to find the value of the fraction.

Handling rational expressions requires careful attention to the values being substituted to avoid undefined expressions, but within this particular exercise, all substitutions are valid, and computation proceeds smoothly.

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence, where each term after the first is obtained by adding a constant difference to the previous term. However, in our exercise, although we deal with a series (summation of terms), it is not an arithmetic series.

The series \(\sum_{k=3}^6 \frac{k-5}{k-1}\), though not arithmetic, involves a finite number of terms which we sum after evaluating each term individually. It's crucial to differentiate between different types of series and sequences for proper understanding and application:

• An arithmetic series has a constant difference between terms.
• Our current series involves a function of \(k\) being summed over several values, which does not follow an arithmetic pattern.

Being familiar with different series types helps in identifying suitable methods for summation and in recognizing the structure of problems involving sequences.