Problem 54 - Precalculus Mathematics for Calculus

Question

A partial sum of an arithmetic sequence is given. Find the sum. $$\sum_{n=0}^{20}(1-2 n)$$

Step-by-Step Solution

Answer

The sum of the sequence is -399.

Step 1: Identify the General Term

The general term of the sequence is given as \(a_n = 1 - 2n\). We will use this to find the sum of the sequence.

Step 2: Use the Formula for Arithmetic Series Sum

The sum of an arithmetic sequence from \(n=0\) to \(n=20\) can be calculated using the formula \( S_n = \frac{n+1}{2} (a + l) \), where \(a\) is the first term and \(l\) is the last term of the sequence.

Step 3: Find the First and Last Terms

The first term \(a\) when \(n=0\) is \(1 - 2(0) = 1\). The last term \(l\) when \(n=20\) is \(1 - 2(20) = -39\).

Step 4: Apply the Formula

Plug \(n=20\), \(a=1\), and \(l=-39\) into the formula: \[ S_{20} = \frac{20+1}{2} (1 + (-39)) \] \[ S_{20} = \frac{21}{2} (-38) \]

Step 5: Calculate the Sum

Now, calculate the sum: \[ S_{20} = \frac{21}{2} \cdot (-38) = 21 \cdot (-19) = -399 \].

Step 6: Conclusion

The sum of the sequence from \(n=0\) to \(n=20\) is \(-399\).

Key Concepts

Partial Sum

In mathematics, the partial sum of an arithmetic sequence represents the sum of a specified number of terms within that sequence. It can be thought of as a way to encapsulate only a portion of the sequence rather than its entire range. This approach is extremely helpful in arithmetic sequences where we're interested in the sum up to a specific term number.
For example, considering the sequence represented by the general term given as \(a_n = 1 - 2n\), the partial sum from \(n=0\) to \(n=20\) can be determined by calculating the sum of each term from the zeroth up to the twentieth term.
Understanding partial sums allows for insight into cumulative behaviors of sequences, especially when each term follows a predictable pattern.

Arithmetic Series Sum Formula

An arithmetic series is a sequence of numbers where the difference between consecutive terms remains consistent. To calculate the sum of these terms efficiently, we use the arithmetic series sum formula. The formula is stated as:
\[ S_n = \frac{n+1}{2} (a + l) \]
where:

• \(S_n\) is the sum of the first \(n+1\) terms of the sequence.
• \(a\) represents the first term.
• \(l\) represents the last term.

This formula is derived by observing that an arithmetic sequence can be paired in a clever way such that the result simplifies into a convenient multiplication structure. By recognizing these pairings, this formula becomes a quick tool for computing series sums in many practical scenarios.

General Term Identification

The general term of an arithmetic sequence is crucial for understanding the entire sequence structure. It describes the formula used to find any term in the sequence by plugging in the term's index number.
For the given sequence, the general term is provided as \(a_n = 1 - 2n\). This indicates how each term in the sequence can be computed sequentially by simply replacing \(n\) with the desired position of the term.
Each term thus follows this linear function, which can reveal how terms grow or diminish across the sequence. Grasping the concept of the general term helps in development and summation of sequences effectively.

First and Last Terms in Sequence

Identifying the first and last terms in an arithmetic sequence is a fundamental step when calculating the sequence's sum. These terms form the basis to apply the arithmetic series sum formula effectively.
For the sequence defined by \(a_n = 1 - 2n\):

• The first term \(a\) is calculated when \(n = 0\): \(a_0 = 1 - 2 \times 0 = 1\).
• The last term \(l\) is identified when \(n = 20\): \(a_{20} = 1 - 2 \times 20 = -39\).

By knowing these specific terms, we can accurately plug them into our sum formula to quickly compute the sum of the sequence's specified range.