Problem 52 - College Algebra with Corequisite Support

Question

Find the smallest value of \(n\) such that \(\sum_{k=1}^{n}(3 k-5)>100 .\)

Step-by-Step Solution

Answer

The smallest value of \( n \) is 10.

Step 1: Understanding the Problem

We need to find the smallest integer \( n \) for which the sum of the sequence \( \sum_{k=1}^{n} (3k - 5) \) exceeds 100.

Step 2: Write the Sum in Sigma Notation

We have the expression \( S_n = \sum_{k=1}^{n} (3k - 5) \). This represents the sum of the arithmetic sequence where each term is defined by \( 3k - 5 \).

Step 3: Find the General Formula for the Sum

We expand the expression \( S_n = \sum_{k=1}^{n} (3k - 5) = \sum_{k=1}^{n} 3k - \sum_{k=1}^{n} 5 \). The first part \( \sum_{k=1}^{n} 3k = 3\sum_{k=1}^{n} k \) represents the sum of the first \( n \) natural numbers multiplied by 3, and \( \sum_{k=1}^{n} 5 = 5n \) as each 5 is added \( n \) times.

Step 4: Calculate the Sum Formula

The sum of the first \( n \) natural numbers is \( \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \). Substitute this into the formula to get \( S_n = 3 \times \frac{n(n+1)}{2} - 5n = \frac{3n(n+1)}{2} - 5n \).

Step 5: Simplify the Sum Expression

We simplify \( \frac{3n(n+1)}{2} - 5n = \frac{3n^2 + 3n - 10n}{2} = \frac{3n^2 - 7n}{2} \). Hence, the sum expression is \( S_n = \frac{3n^2 - 7n}{2} \).

Step 6: Set the Inequality for the Sum

We want \( S_n > 100 \), so we set up the inequality \( \frac{3n^2 - 7n}{2} > 100 \). Multiply both sides by 2 to clear the fraction: \( 3n^2 - 7n > 200 \).

Step 7: Solve the Quadratic Inequality

We solve the quadratic inequality \( 3n^2 - 7n - 200 > 0 \). First, solve the equation \( 3n^2 - 7n - 200 = 0 \) using the quadratic formula: \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a=3, b=-7, c=-200 \).

Step 8: Calculate Discriminant

Calculate the discriminant: \( b^2 - 4ac = (-7)^2 - 4 \times 3 \times (-200) = 49 + 2400 = 2449 \).

Step 9: Finish Solving the Quadratic

Substitute back to find \( n \): \( n = \frac{7 \pm \sqrt{2449}}{6} \). Calculate \( \sqrt{2449} \), which is approximately 49.49, giving solutions \( n = \frac{7 + 49.49}{6} \approx 9.915 \) and \( n = \frac{7 - 49.49}{6} \approx -7.915 \).

Step 10: Determine the Smallest Integer Value

Since \( n \) must be a positive integer, and \( n \approx 9.915 \), the smallest integer satisfying this inequality is \( n = 10 \).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers in which the difference between consecutive terms remains constant. This difference is called the 'common difference'. An example of such a sequence might be: 2, 5, 8, 11, and so forth, where the common difference is 3. It's essential to recognize that every term after the first is obtained by adding the common difference to the preceding term. If we denote the first term by \( a \) and the common difference by \( d \), the \( n^{th} \) term \( a_n \) is given by:

• \( a_n = a + (n-1) \, d \)

The sum of the first \( n \) terms is also straightforward and can be derived by multiplying the number of terms by the average of the first and last terms. The formula for the sum \( S_n \) of the first \( n \) terms in an arithmetic sequence is:

• \( S_n = \frac{n}{2} \times (a + a_n) \)

Understanding arithmetic sequences simplifies many math problems, as they frequently occur in number patterns, real-life scenarios, and even in solving quadratic inequalities as seen in the given exercise.

Sigma Notation

Sigma notation, represented by the Greek letter \( \Sigma \), is a concise and efficient way to express the sum of a sequence. It is particularly useful in mathematics because it allows us to write sums with many terms without listing each term.
When you see an expression like \( \sum_{k=1}^{n}(3k-5) \), it can be broken down as follows:

• \( k \): the index of summation, starting from \( 1 \) and ending at \( n \)
• \( 3k-5 \): the expression for each term, dependent on \( k \)

Each value of \( k \) from 1 to \( n \) is substituted into the expression \( 3k-5 \) and the results are summed. \Sigma notation becomes particularly powerful when dealing with large series and organizing them into manageable expressions, especially involving sums like those in arithmetic sequences.

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). This formula provides the solutions, also known as the roots, of any quadratic equation and is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, "\( \pm \)" indicates there are usually two solutions, corresponding to the addition and subtraction of the square root term.
The formula is derived from the process of completing the square and gives exact solutions for any quadratic equation. In our context of quadratic inequalities, it helps us determine critical points where the expression changes sign, thus allowing us to find intervals satisfying particular conditions like inequality signs.
When dealing with quadratic inequalities, such as \( 3n^2 - 7n - 200 > 0 \), solving the equation \( 3n^2 - 7n - 200 = 0 \) gives us threshold points around which we explore the inequality.

Discriminant

The discriminant is a part of the quadratic formula found under the square root sign: \( b^2 - 4ac \). It plays a crucial role in determining the nature of the roots of a quadratic equation.
Depending on its value, the characteristics of the solutions can be different:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root, meaning the roots are equal, providing a perfect square trinomial.
• If the discriminant is negative, it results in complex roots (imaginary numbers).

In the context of solving inequalities, the discriminant is used to check whether the solutions are real numbers, which in turn affects how we interpret the quadratic inequality outcomes. For the exercise, calculating \( b^2 - 4ac = 2449 \) established that the roots were real and distinct, enabling further problem solving within that realm.