Problem 54 - Algebra and Trigonometry

Question

\(49-54\) . 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 Given Formula

The given expression represents a partial sum of the arithmetic sequence defined by the equation \( a_n = 1 - 2n \). This formula will be used to identify the first term, the common difference, and the number of terms.

Step 2: Determine the First Term and the Common Difference

Substitute \( n = 0 \) into the formula \( a_n = 1 - 2n \) to find the first term, \( a_0 = 1 - 2(0) = 1 \). The common difference \( d \) can be found by checking the difference between the first two terms: \( a_1 = 1 - 2(1) = -1 \). Thus, the common difference \( d = -1 - 1 = -2 \).

Step 3: Determine the Number of Terms

Since we need to sum the sequence from \( n = 0 \) to \( n = 20 \), the sequence contains \( 20 - 0 + 1 = 21 \) terms.

Step 4: Calculate the Last Term

To find the last term of the sequence, substitute \( n = 20 \) into the formula \( a_{20} = 1 - 2(20) = 1 - 40 = -39 \).

Step 5: Use the Formula for the Sum of an Arithmetic Sequence

The sum \( S_n \) of the first \( n \) terms of an arithmetic sequence is given by the formula: \[ S_n = \frac{n}{2} (a_0 + a_n) \] where \( a_0 \) is the first term and \( a_n \) is the last term.

Step 6: Substitute the Values into the Formula

Substitute \( n = 21 \), \( a_0 = 1 \), and \( a_{20} = -39 \) into the formula: \[ S_{21} = \frac{21}{2} (1 + (-39)) = \frac{21}{2} (-38) \].

Step 7: Calculate the Result

Simplify the expression \( \frac{21}{2} \times -38 \) and calculate the result: \[ S_{21} = 21 \times -19 = -399 \]. Thus, the sum of the sequence is \(-399\).

Key Concepts

Partial Sum

A partial sum is the sum of a specified number of terms in a sequence. In this case, we are looking at the partial sum of an arithmetic sequence. This sequence is defined by the formula \( a_n = 1 - 2n \). The problem requires us to find the sum of the terms from \( n = 0 \) to \( n = 20 \). This involves calculating the sum of the first 21 terms of the sequence. To compute the partial sum, we need to know several key components of the sequence such as the first term, the last term, and the number of terms in the sequence.

Common Difference

The common difference in an arithmetic sequence is the consistent amount that each term increases or decreases from the previous term. It can be considered the 'step' between terms. For the given problem, the arithmetic sequence is defined by \( a_n = 1 - 2n \). To find the common difference \( d \), we can subtract the first term from the second term.

• First term (\( a_0 \)): \( 1 \)
• Second term (\( a_1 \)): \( -1 \)

Thus, \( d = a_1 - a_0 = -1 - 1 = -2 \). This means that each subsequent term is 2 less than the previous one.

Sum of Arithmetic Sequence

The sum of an arithmetic sequence can be found using a straightforward formula. When given

• First term \( a_0 \)
• Last term \( a_n \)
• Number of terms \( n \)

We use the formula: \[ S_n = \frac{n}{2} (a_0 + a_n) \]For this problem, applying the formula involves substituting the values:

• First term, \( a_0 = 1 \)
• Last term, \( a_{20} = -39 \)
• Number of terms, \( n = 21 \)

Substitute into the formula: \[ S_{21} = \frac{21}{2} (1 + (-39)) \]Simplifying gives you \[ S_{21} = \frac{21}{2} \times (-38) = 21 \times -19 = -399 \]So, the sum of the sequence is \(-399\).

Number of Terms

The number of terms in a sequence is simply the count of all the terms included in the partial sum. For this problem, we're asked to find the sum from \( n = 0 \) through \( n = 20 \), and this involves counting all those terms. The calculation is straightforward:

• Start at term \( n = 0 \)
• End at term \( n = 20 \)

The number of terms \( n \) is given by:\[ n = 20 - 0 + 1 = 21 \]Understanding that we have 21 terms helps us apply the sum of sequence formulas effectively. By knowing the number of terms, it becomes easy to substitute into the formula for the sum of an arithmetic sequence, ensuring accurate results.