Problem 50 - Precalculus Mathematics for Calculus

Question

A partial sum of an arithmetic sequence is given. Find the sum. $$-3+\left(-\frac{3}{2}\right)+0+\frac{3}{2}+3+\cdots+30$$

Step-by-Step Solution

Answer

The sum of the sequence is 310.5.

Step 1: Identify Sequence Characteristics

We are given the arithmetic sequence: \[-3, -\frac{3}{2}, 0, \frac{3}{2}, 3, \ldots, 30\].First, identify the first term \(a\) and the common difference \(d\). The first term \(a = -3\). To find the common difference \(d\), subtract the first term from the second: \(-\frac{3}{2} - (-3) = \frac{3}{2}\). So, \(d = \frac{3}{2}\).

Step 2: Determine the Number of Terms

The last term given is \(30\). Use the nth-term formula for an arithmetic sequence \(a_n = a + (n-1)d\) to find \(n\). Substitute into the formula:\[30 = -3 + (n-1)\cdot\frac{3}{2}\]Solving for \(n\), we get:\[33 = (n-1)\cdot\frac{3}{2}\]\[(n-1) = \frac{66}{3} = 22\]\[n = 23\]

Step 3: Calculate the Sum

Use the formula for the sum of an arithmetic sequence:\[S_n = \frac{n}{2} \cdot (a + l)\]where \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term.\[S_{23} = \frac{23}{2} \cdot (-3 + 30)\]Calculate:\[S_{23} = \frac{23}{2} \cdot 27\]\[S_{23} = \frac{621}{2} = 310.5\]

Step 4: Conclude the Calculation

Verify that the formula and steps are correct, and interpret the final result. The calculated sum of this arithmetic sequence from -3 to 30, is \(310.5\).

Key Concepts

Partial Sum

The concept of a "partial sum" in an arithmetic sequence refers to the sum of a specified number of terms, starting from the first term and ending at a certain term in the sequence. In practice, calculating a partial sum allows us to determine the total value from one point to another in the sequence.
To find the partial sum, we use the formula: \[ S_n = \frac{n}{2} \cdot (a + l) \] where

• \( S_n \) is the partial sum,
• \( n \) is the number of terms,
• \( a \) is the first term, and
• \( l \) is the last term in the sequence involved in the sum.

To solve an exercise like the one provided, understanding partial sums helps you to quickly compute the result without individually adding each term.
It simplifies handling sequences by using the properties of arithmetic sequences to calculate the sum efficiently.

Common Difference

The "common difference" is a fundamental characteristic of an arithmetic sequence. It is the constant amount that we add or subtract to one term in order to obtain the next term.
In an arithmetic sequence like \[ -3, -\frac{3}{2}, 0, \frac{3}{2}, 3, \ldots \],we find the common difference \( d \) by subtracting any term from its subsequent term. For instance, \[ d = -\frac{3}{2} - (-3) = \frac{3}{2}.\]
This consistent difference indicates the sequence's progression.
Recognizing this common difference is crucial for projecting and understanding the sequence's pattern.
It plays a pivotal role in determining other sequence properties like the nth term and the sum of terms.

First Term

The "first term" in an arithmetic sequence is the starting point from where the sequence begins. It is denoted by \( a \).
In our example, the first term is \( a = -3 \).
Knowing the first term is essential due to its influence on both the determination of the nth term and the calculation of partial sums.
It acts as the starting reference point and helps in developing the entire sequence based on the common difference.
Without the first term, forming subsequent sequence terms and performing calculations would be challenging.

Number of Terms

The "number of terms" in an arithmetic sequence denotes how many terms, starting from the first, are included in the sequence or a partial sequence.
To find it, particularly in problems like the given one, we use the nth-term formula for arithmetic sequences:\[ a_n = a + (n-1)d \].
Here, we solve for \( n \), the number of terms. For the sequence ending at 30:\[ 30 = -3 + (n-1)\cdot\frac{3}{2} \].
Solving this provides us with the value of \( n = 23 \).
This calculation is essential for employing the sum formula because \( n \) represents the upper limit of terms summed. Understanding how to find \( n \) helps complete arithmetic sequence problems confidently.