Problem 62 - Precalculus: Mathematics for Calculus

Question

A partial sum of an arithmetic sequence is given. Find the sum. $$-10-9.9-9.8-\dots-0.1$$

Step-by-Step Solution

Answer

The sum of the sequence is -505.

Step 1: Identify the Elements of the Arithmetic Sequence

First, identify the key elements of the arithmetic sequence, which are the first term (\(a_1\)), the last term (\(a_n\)), and the common difference (\(d\)). The given sequence starts at \(-10\) and decreases to \(-0.1\) with a common difference \(d = 0.1\).

Step 2: Find the Number of Terms

To find the number of terms (\(n\)), use the formula for the \(n\)-th term of an arithmetic sequence: \( a_n = a_1 + (n-1) \cdot d \).Substitute \(a_n = -0.1\), \(a_1 = -10\), and \(d = 0.1\) into the formula:\(-0.1 = -10 + (n-1) \cdot 0.1\).Solve for \(n\): \((n-1) \cdot 0.1 = 9.9\) so \(n - 1 = 99\), and therefore, \(n = 100\).

Step 3: Use the Sum Formula for an Arithmetic Sequence

The sum (\(S_n\)) of an arithmetic sequence can be found using the formula: \[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \].Substitute \(n=100\), \(a_1=-10\), and \(a_n=-0.1\) into the formula:\[ S_{100} = \frac{100}{2} \cdot (-10 + (-0.1)) = 50 \cdot (-10.1) = -505 \].

Key Concepts

Sum of Arithmetic Sequence

Understanding the sum of an arithmetic sequence is crucial in mathematics. An arithmetic sequence is a series of numbers in which each term differs from the previous one by a constant known as the common difference. To calculate the sum of an arithmetic sequence, we apply a formula that considers both the number of terms and the first and last terms of the sequence.

In general, the formula for finding the sum of an arithmetic sequence is:

• \(S_n = \frac{n}{2} \cdot (a_1 + a_n)\)

This equation elegantly captures the sum by multiplying the average of the first and last terms by the number of terms. It works effectively because arithmetic sequences maintain a uniform pattern of increments or decrements, allowing for predictable computation of total sums.

By using this principle, even complex sequences like the one given in the exercise (\(-10-9.9-9.8-\dots-0.1\) ) can be tackled with ease.

Common Difference in Arithmetic Sequence

The common difference in an arithmetic sequence is the key characteristic that defines the sequence. It is the consistent interval between consecutive terms. In simpler terms, it's what you add or subtract to go from one number in the sequence to the next. If a sequence has a common difference, it will change systematically, either increasing or decreasing.

To calculate the common difference, one can simply subtract any term from the subsequent term in the sequence. Let’s consider our example, where the first term (\(a_1\)) is \(-10\) and each term increases by \(0.1\). Hence, the common difference \(d\) is \(0.1\) .

This constant value, the common difference, ensures we stay on the correct path within the sequence, allowing us to predict future terms and, ultimately, calculate extended properties like the sequence's sum.

Number of Terms in Sequence

Determining the number of terms in an arithmetic sequence is an important step when dealing with such sequences. The number of terms, denoted by \(n\), is essential for summing processes or understanding the sequence's structure.

To find the number of terms, use the formula for the n-th term of an arithmetic sequence:

• \(a_n = a_1 + (n-1) \cdot d\)

For the sequence here, where \(a_n = -0.1\), \(a_1 = -10\) and \(d = 0.1\) , substitute into the formula to solve for \(n\).

It leads to:

• \(-0.1 = -10 + (n-1) \cdot 0.1\)
• \(n-1 = 99\)
• \(n = 100\)

This process of calculating \(n\) ensures we comprehend the sequence length accurately, enabling easier computation for further analysis like the sequence's total sum or any term's position.