Problem 48 - Precalculus: Mathematical 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 first term and common difference

The sequence given is arithmetic, with the first term \( a_1 = -10 \). We can see that each term increases by 0.1. Therefore, the common difference \( d \) is 0.1.

Step 2: Find the number of terms in the sequence

To find the number of terms \( n \), we use the last term given in the sequence, which is \( -0.1 \). Using the formula for the \( n \)-th term of an arithmetic sequence \( a_n = a_1 + (n-1)d \), we substitute to get: \[ -0.1 = -10 + (n-1)(0.1) \] Solve for \( n \):\[ 0.1n - 0.1 = n \] \[ -0.1 = -10 + 0.1n - 0.1 \]\[ 0.1n = 9.9 \]\[ n = \frac{9.9}{0.1} = 99 \].Therefore, there are 100 terms in the sequence.

Step 3: Use the formula for the sum of an arithmetic sequence

The formula for the sum of the first \( n \) terms of an arithmetic sequence is given by:\[ S_n = \frac{n}{2} (a_1 + a_n) \]Here, \( n = 100, a_1 = -10, a_n = -0.1 \). Substitute these values to find the sum:\[ S_{100} = \frac{100}{2} (-10 + (-0.1)) \]\[ S_{100} = 50 (-10.1) \]\[ S_{100} = -505 \].

Key Concepts

Partial Sum

A partial sum is the sum of a specified number of sequential terms from a given sequence. When dealing with arithmetic sequences, the partial sum is the accumulation of a set number of terms. Imagine you are gradually adding each term of the sequence one by one. With each addition, you create a new sum, which is larger than the sum of all previous terms. In essence, each incremental calculation provides part of the whole sum. In our case, we determined the sum of the first 100 terms of the arithmetic sequence. This type of exercise helps you understand how partial sums can build up considerably by adding individual, even small, numbers like 0.1 in our problem.

Common Difference

In any arithmetic sequence, the common difference is the consistent interval between consecutive terms. It dictates how the sequence grows (or diminishes) as you move from one term to the next. In the exercise, each subsequent number increases by 0.1, which serves as the common difference, denoted by the symbol \( d \). The knowledge of the common difference is crucial because this value allows you to identify missing numbers within a sequence. It also plays a foundational role when deriving formulas related to arithmetic sequences. Recognizing this difference is your first step in understanding and mastering arithmetic progressions.

Number of Terms

Figuring out the number of terms in a sequence is often an important step when calculating the sum of that sequence. In our exercise, the sequence starts at -10 and concludes at -0.1. To determine how many terms your sequence has, you utilize the formula for the \( n \)-th term, \( a_n = a_1 + (n-1)d \). By plugging in the known terms and solving for \( n \), you reveal the total number of terms present. In the original problem, the solution indicates that there are 100 terms. This resulted from solving the equation \( -0.1 = -10 + (n-1)(0.1) \). Understanding the number of terms is crucial for determining the complete sum of a sequence, as it defines the actual content of your arithmetic progression.

Sum Formula

The formula to find the sum of an arithmetic sequence is a powerful tool in mathematics. It is given by \( S_n = \frac{n}{2} (a_1 + a_n) \). This succinct formula allows you to place all relevant values at once to discover the complete sum of your chosen terms. In our scenario, \( n \), the number of terms, was determined to be 100, while \( a_1 = -10 \) and \( a_n = -0.1 \) represent the first and last terms, respectively. By substituting these into the sum formula, you calculate the entire sequence's sum concisely as \( S_{100} = -505 \). This formula is tremendously useful as it provides a shortcut to finding sums without manually adding each term one-by-one, especially when dealing with large sequences.