Problem 52 - 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 Sequence Parameters

An arithmetic sequence is defined by its first term, common difference, and number of terms. The first term is given as \( a = -10 \). The common difference \( d \) between consecutive terms can be found by subtracting the first term from the second term: \( d = -9.9 - (-10) = 0.1 \).

Step 2: Determine the Number of Terms

To find the number of terms \( n \), use the formula for the n-th term of an arithmetic sequence: \( a_n = a + (n-1) \cdot d \). The last term is \( a_n = -0.1 \). Substitute into the equation: \(-0.1 = -10 + (n-1) \cdot 0.1 \). Solve for \( n \): $$\begin{align*} -0.1 &= -10 + (n-1) \cdot 0.1 \ 9.9 &= (n-1) \cdot 0.1 \ n-1 &= 99 \ n &= 100 \end{align*}$$ So, there are 100 terms.

Step 3: Calculate the Sum of the Sequence

Use the formula for the sum of an arithmetic sequence: \( S_n = \frac{n}{2} \cdot (a + a_n) \). Substitute the known values: \( a = -10 \), \( a_n = -0.1 \), and \( n = 100 \). Calculate the sum: \[ S_{100} = \frac{100}{2} \cdot (-10 + (-0.1)) = 50 \cdot (-10.1) = -505 \] So, the sum of the sequence is \(-505\).

Key Concepts

Sum of Sequence

The sum of an arithmetic sequence can be calculated easily if you know some key parameters, such as the number of terms, the first term, and the last term. For our arithmetic sequence, the sum is calculated using the formula:

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

Here, \( S_n \) is the sum of the sequence, \( n \) is the number of terms, \( a \) is the first term, and \( a_n \) is the last term.
Plug these numbers into your formula and you'll find the sum in no time! For our example, the sequence from \(-10\) to \(-0.1\) with 100 terms has a sum of \(-505\). This method is useful, particularly when dealing with long sequences, as it eliminates the need to add each term manually.

Common Difference

In arithmetic sequences, the 'common difference' is what defines the pattern of the sequence. This difference is constant between any two consecutive terms. To find the common difference \( d \), you subtract any term from the term that follows it.
For example, in the sequence \(-10, -9.9, -9.8, \ldots\), the common difference is:

• \( d = -9.9 - (-10) = 0.1 \)

This small positive number shows the increase from one term to the next, producing our arithmetic sequence. Getting the hang of finding the common difference is crucial, as it helps you define and work with any arithmetic sequence.

nth Term Formula

The nth term formula is a crucial tool to find any term in an arithmetic sequence. The formula goes as follows:

• \( a_n = a + (n-1) \cdot d \)

Where \( a_n \) is the nth term, \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference.
In our example, \( a = -10 \), and \( d = 0.1 \). Suppose you want to find the 50th term, you'd plug the numbers in:

• \( a_{50} = -10 + (50-1) \cdot 0.1 = -10 + 4.9 = -5.1 \)

This formula is indispensable when you need to find a specific term without listing the entire sequence.

Total Number of Terms

Determining the total number of terms in an arithmetic sequence is another important step. This involves using the formula for the nth term, but in reverse. You're solving for \( n \):
Using our formula, identify the last term as \( a_n = -0.1 \), and solve:

• \( -0.1 = -10 + (n-1) \cdot 0.1 \)
• Solve for \( n \): \[ \begin{align*}-0.1 &= -10 + (n-1) \cdot 0.1 \9.9 &= (n-1) \cdot 0.1 -1 &= 99 &= 100\end{align*} \]

This shows that there are 100 terms in our sequence. Knowing how many terms you have is essential for applying the sum formula effectively.