Problem 50 - Algebra and Trigonometry

Question

\(49-54\) . 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 the sequence properties

The given sequence is arithmetic because each term increases by a constant amount. The first term \( a_1 = -3 \) and the sequence looks like \(-3 + \left(-\frac{3}{2}\right) + 0 + \frac{3}{2} + 3 + \cdots + 30\). The sequence increases by \(\frac{3}{2}\), which is the common difference \(d\).

Step 2: Calculate the number of terms

The last term given in the sequence is 30. We can use the formula for the \(n\)th term of an arithmetic sequence: \( a_n = a_1 + (n - 1) \cdot d \). Rearranging for \(n\), we have: \( n = \frac{a_n - a_1}{d} + 1 \). Substituting the known values \( a_n = 30 \), \( a_1 = -3 \), and \( d = \frac{3}{2} \): \( n = \frac{30 - (-3)}{\frac{3}{2}} + 1 = \frac{33}{\frac{3}{2}} + 1 = \frac{33 \cdot 2}{3} + 1 = 22 + 1 = 23 \). So, there are 23 terms in the sequence.

Step 3: Use the sum formula for arithmetic sequences

The formula for the sum of \(n\) terms of an arithmetic sequence is \( S_n = \frac{n}{2} \cdot (a_1 + a_n) \). Substituting the values: \( n = 23 \), \( a_1 = -3 \), \( a_n = 30 \), the sum is \( S_{23} = \frac{23}{2} \cdot (-3 + 30) = \frac{23}{2} \cdot 27 = 23 \cdot 13.5 = 310.5 \).

Key Concepts

Partial Sum

In an arithmetic sequence, a partial sum refers to the total sum of a certain number of terms from the beginning of the sequence up to a specified point or term. Understanding this concept is crucial because it allows us to calculate the total value of a part of the sequence without having to add each term individually.
For instance, if we have a sequence like \(-3, -\frac{3}{2}, 0, \frac{3}{2}, 3, \ldots, 30\), the partial sum could represent the sum of the first 23 terms. This means calculating the sum from \(-3\) to \(30\).
Understanding how to find partial sums can be highly beneficial when dealing with long sequences. Instead of adding each term separately, we use an arithmetic sequence sum formula to quickly find the answer, saving time and effort. This concept is particularly useful in mathematics and practical applications where adding many numbers is necessary.

Common Difference

The common difference in an arithmetic sequence is the constant amount that each term increases or decreases by compared to the previous term. It is what makes the sequence 'arithmetic' as opposed to 'geometric' or any other kind of sequence.
In the sequence \(-3, -\frac{3}{2}, 0, \frac{3}{2}, 3, \ldots\), the common difference is \(\frac{3}{2}\). This means that each term is obtained by adding \(\frac{3}{2}\) to the previous term. For example:

• Adding \(\frac{3}{2}\) to \(-3\) gives \(-\frac{3}{2}\).
• Continuing this way eventually leads to the last term in the series, which is 30.

Recognizing the common difference is essential for identifying the pattern in an arithmetic sequence and being able to calculate other important aspects, like partial sums or the number of terms.

Arithmetic Sequence Sum Formula

The arithmetic sequence sum formula is a powerful tool for finding the total sum of a sequence without adding each term individually. The formula is given by:
\[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \]
where:
- \(S_n\) is the sum of the first \(n\) terms
- \(n\) is the number of terms
- \(a_1\) is the first term
- \(a_n\) is the nth term
In our example, with a sequence starting at \(-3\) and ending at \(30\), and having 23 terms, we used this formula to find the sum:

• Substitute \(n = 23\), \(a_1 = -3\), and \(a_n = 30\).
• Perform the calculations: \( S_{23} = \frac{23}{2} \cdot (-3 + 30) = 310.5 \).

This method streamlines the process of adding several terms and is particularly helpful in academic settings where arithmetic sequences are common.

Number of Terms in Sequence

Knowing the number of terms in an arithmetic sequence is critical for various calculations, like finding the partial sum. To determine the number of terms, you can use the formula for the nth term of an arithmetic sequence:
\[ a_n = a_1 + (n - 1) \cdot d \]
By rearranging, we'll find \(n\):
\[ n = \frac{a_n - a_1}{d} + 1 \]
Let's break it down with our example where:

• \(a_n\) is the last term, which is 30.
• \(a_1\) is the first term, which is \(-3\).
• \(d\), the common difference, is \(\frac{3}{2}\).

Substituting these values, we calculate:
\[ n = \frac{30 - (-3)}{\frac{3}{2}} + 1 = 23 \]
This shows that the sequence contains 23 terms. By identifying the number of terms effectively, you make it easier to perform other calculations, like finding those partial sums with accuracy.