Problem 32 - College Algebra with Modeling and Visualization

Question

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{1}=6, a_{5}=-30 $$

Step-by-Step Solution

Answer

The sum of the first 20 terms is -1590.

Step 1: Understand the problem

You need to find the sum of the first 20 terms of an arithmetic sequence given that the first term \(a_1 = 6\) and the fifth term \(a_5 = -30\).

Step 2: Identify the common difference

Use the formula for the \(n\)-th term of an arithmetic sequence: \(a_n = a_1 + (n-1) imes d\). Using \(a_5 = -30\), we have: \(-30 = 6 + 4d\). Solve for \(d\):\[-30 = 6 + 4d \-36 = 4d \d = -9\]

Step 3: Calculate the 20th Term

To use the formula for the sum of the sequence, first find \(a_{20}\) using \(a_n = a_1 + (n-1) imes d\) with \(n = 20\):\[a_{20} = 6 + (20-1) imes (-9) \a_{20} = 6 - 171 \a_{20} = -165\]

Step 4: Use the sum formula for an arithmetic sequence

The formula for the sum of the first \(n\) terms of an arithmetic sequence is \(S_n = \frac{n}{2} imes (a_1 + a_n)\). Use this to find \(S_{20}\):\[S_{20} = \frac{20}{2} imes (6 + (-165)) \S_{20} = 10 imes (-159) \S_{20} = -1590\]

Step 5: Verify your steps

Ensure there are no calculation errors by rechecking each step from determining \(d\) to using the sum formula. The common difference is consistent in the calculations and the final sum matches expectations.

Key Concepts

Common Difference

The common difference in an arithmetic sequence is the difference between any two successive terms. It's a constant value, meaning that every term increases or decreases by the same amount. This is what makes the sequence "arithmetic." In our example, to find the common difference \(d\), we use the given terms:

• First term \(a_1 = 6\)
• Fifth term \(a_5 = -30\)

To find \(d\), use the formula for the \(n\)-th term: \(a_n = a_1 + (n-1)\times d\). Plugging in the values for \(a_5\), we get \(-30 = 6 + 4d\). Solving for \(d\), we find that \(d = -9\). This tells us each term is 9 less than the previous one.

Sum of Sequence

The sum of an arithmetic sequence is a key operation you might need to perform. It lets you find the total of the first \(n\) terms without adding them individually. This is especially useful for long sequences.The formula used here is \(S_n = \frac{n}{2} \times (a_1 + a_n)\). This requires knowing the first term \(a_1\) and the \(n\)-th term \(a_n\).In the exercise, to find the sum of the first 20 terms \(S_{20}\), we have:

• \(a_1 = 6\)
• \(a_{20} = -165\)

So, \(S_{20} = 10 \times (6 + (-165)) = 10 \times (-159) = -1590\). The sequence sum is \(-1590\).

Formula for nth Term

The formula for the \(n\)-th term is essential for understanding arithmetic sequences. It allows you to find any term in the sequence without listing all terms.The formula is: \(a_n = a_1 + (n-1)\times d\), where:

• \(a_1\) is the first term.
• \(d\) is the common difference.

For \(a_{20}\), given \(a_1 = 6\) and \(d = -9\), the calculation is \(a_{20} = 6 + 19 \times (-9) = 6 - 171 = -165\). This provides the 20th term as \(-165\).

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. Understanding this concept helps in dealing with problems that involve summing elements of such sequences.When working with an arithmetic series, you often use the sum formula \(S_n = \frac{n}{2} \times (a_1 + a_n)\) to quickly calculate the total. This combines the straightforward nature of adding a known series with the efficiency needed for larger numbers.In the given problem, the arithmetic series was summed using 20 terms, resulting in \(-1590\). This is a practical demonstration of how powerful the arithmetic series formula is for solving sequence sum problems efficiently.